Vapor pressure Monte Carlo simulation

Fig. 2. Adsorption isotherms in all four model systems. F is the Gibbs excess adsoiption. The pressures corresponding to the three configurations shown in Figure 3 are marked with arrows. The pressure is plotted relative to the vapor pressure of the model fluid, as determined by independent Gibbs Ensemble Monte Carlo simulations. Chemical potentials were converted to pressures using a virial equation of state.

In Fig. 8.2, the points are the results obtained by evaluating the transverse component of the pressure tensor obtained from Monte Carlo simulation and the line is the result of an integration of the Gibbs adsorption isotherm using the simulation data. The vertical dashed line shows the saturation vapor pressure. 

Figure 4. Reversible work of formation of a physically consistent cluster as a function of the number of molecules in the cluster, /, and the cluster volume, v. Monte Carlo simulation for supercooled Ar vapor (Lennard-Jones potential) at 70 K and various pressures [67, 5].

Two simulation methods—Monte Carlo and molecular dynamics—allow calculation of the density profile and pressure difference of Eq. III-44 across the vapor-liquid interface [64, 65]. In the former method, the initial system consists of N molecules in assumed positions. An intermolecule potential function is chosen, such as the Lennard-Jones potential, and the positions are randomly varied until the energy of the system is at a minimum. The resulting configuration is taken to be the equilibrium one. In the molecular dynamics approach, the N molecules are given initial positions and velocities and the equations of motion are solved to follow the ensuing collisions until the set shows constant time-average thermodynamic properties. Both methods are computer intensive yet widely used. 

The shape of a zeolite sorption uptake isotherm, a quantitation of the amount of a given sorbate taken up as a function of its partial pressure in the gas phase in equilibiitun with the zeolite sorbent, depends both on the zeolite sorbate interaction and on the sorbate - sorbate interactions. Simulation of such isotherms for one or more sorbates is accomplished by the Grand Canonical Monte Carlo method. Additional to the molecular reorientation and movement attempts is a particle creation or annihilation, the probability of which scales with the partial pressure [100,101]. This procedure thus simulates the eqmlibrium between the sorbed phase in the zeolite and an infinite gas / vapor bath. Reasonable reproduction of uptake isotherms for simple gases has been achieved for a small number of systems (e.g. [100,101]), and the molecular simulations have, for example, explained at a molecular level the discontinuity observed in the Ar - VPI-5 isotherm. 

Molecular Simulations Molecular simulations are useful for predicting properties of bulk fluids and solids. Molecular dynamics (MD) simulations solve Newton s equations of motion for a small number (on the order of 10 ) of molecules to obtain the time evolution of the system. MD methods can be used for equilibrium and transport properties. Monte Carlo (MC) simulations use a model for the potential energy between molecules to simulate configurations of the molecules in proportion to their probability of occurrence. Statistical averages of MC configurations are useful for equilibrium properties, particularly for saturated densities, vapor pressures, etc. Property estimations using molecular simulation techniques are not illustrated in the remainder of this section as commercial software implementations are not generally available at this time. 

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]. 

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