The Grand Canonical Monte Carlo Method

One of the most appealing characteristics of the grand canonical Monte Carlo method is that, as in many experimental situations, the chemical potential jx is one of the independent variables. This is the case of low-sweep-rate voltammetry, an electrochemical technique where the electrode potential can be used to control the chemical potential of species at the metal-solution interface. This technique offers a straightforward way of obtaining the adsorption 

The 2D adsorption system is taken as a square lattice with M = Ly. L sites (in the present simulations, we will take L = 100). Each adsorption site is labeled by index 0, 1, or 2, depending on whether it is empty, occupied by one atom of the same kind as those of the substrate, or occupied by one adsorbate atom, respectively. 

Following the procedure proposed by Metropolis and coworkers, the acceptance probability for a transition from state if to if is defined as 

In the grand canonical Monte Carlo simulation, three different types of events are allowed for, as follows  

Adsorption of an adsorbate atom onto a randomly selected lattice site 

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At a fixed T and for a given value of p, the adsorption process has been simulated by using the grand canonical Monte Carlo method [S]. At any elementary step, a site chosen at random is tested to change its occupancy state according to the Metropolis scheme of probabilities where Hf andtf/ are the hamiltonians... 

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. 

In the grand-canonical Monte Carlo method, the system volume, temperature, and chemical potential are kept fixed, while the number of particles is allowed to fluctuate.There exist three types of trial move (1) displacement of a particle, (2) insertion of a particle, and (3) removal of a particle. These trial moves are generated at random with equal probability. The acceptance probability of the Metropolis method can be used for the trial moves of type (1). For the two other types, the acceptance probabilities are different. Regarding zeolites, an adsorption isotherm can be calculated with the grand-canonical Monte Carlo method by running a series of simulations at varying chemical potentials. 

Computer simulations of the isotherm of adsorption are frequently based on the grand canonical Monte Carlo method [40]. If the atomic structure of the solid adsorbent is known and if the adsorbate/solid and adsorbate/adsorbate energies can be computed, this algorithm makes it possible to calculate isotherms of adsorption directly. 

The key feature about the grand canonical Monte Carlo method is that the number of particles may change during the simulation. There are three basic moves in a grand canonical Monte Carlo simulation ... 

Cerius2 (MSI Inc.) was used diroughout the simulations. Forcefield parameters obtained by Mellot et all. [3] are listed in Table I. The Grand Canonical Monte Carlo method (under constant chemical potential (p), volume (V), temperature (T)) was used to get the equilibrium amount adsorbed. 

Molecular parameters used in the above formulation are usually taken fi-om Ref 49. The grand-canonical Monte Carlo method of Adams [50] is commonly used in simulation. 

One application of the grand canonical Monte Carlo simulation method is in the study ol adsorption and transport of fluids through porous solids. Mixtures of gases or liquids ca separated by the selective adsorption of one component in an appropriate porous mate The efficacy of the separation depends to a large extent upon the ability of the materit adsorb one component in the mixture much more strongly than the other component, separation may be performed over a range of temperatures and so it is useful to be to predict the adsorption isotherms of the mixtures. 

To conclude, the introduction of species-selective membranes into the simulation box results in the osmotic equilibrium between a part of the system containing the products of association and a part in which only a one-component Lennard-Jones fluid is present. The density of the fluid in the nonreactive part of the system is lower than in the reactive part, at osmotic equilibrium. This makes the calculations of the chemical potential efficient. The quahty of the results is similar to those from the grand canonical Monte Carlo simulation. The method is neither restricted to dimerization nor to spherically symmetric associative interactions. Even in the presence of higher-order complexes in large amounts, the proposed approach remains successful. 

The arrows indicate a semi-permeable membrane and the species allowed to permeate is shown within the arrows. The parentheses show a GEMC phase (or region) and the species it contains. The first and the last region are also connected to each other. Using such a scheme, Bryk et al. showed that osmotic Monte Carlo can be successfully used to study the association of two different molecular species when an associating intermolecular potential is included in the simulation. The results agreed well with the more traditional grand-canonical Monte Carlo methods. 

The grand canonical ensemble is appropriate for adsorption systems, in which the adsorbed phase is in equilibrium with the gas at some specified temperature. The use of a computer simulation allows us to calculate average macroscopic properties directly without having to explicitly calculate the partition function. The grand canonical Monte Carlo (GCMC) method as applied in this work has been described in detail earlier (55). The aspects involving binary fluid mixtures have been described previously in our Xe-Ar work (30). 

The absolute adsorption isotherm as a function of gas-phase fuga ity is obtained directly from molecular simulations based on the grand canonical Monte Carlo (GCMC) method. Since the difference between absolute and excess adsorption is negligible at sub-atmospheric pressure, the low-pressure portion of the absolute isotherm can adso be determined from experiment. Eq. (2) is suitable for extrapolating the absolute isotherm from low to high pressure and Eq. (3) provides the conversion to excess adsorption. Experiments are needed to test these predictions of adsorption at high pressure. 

The grand canonical Monte Carlo (GCMC) method was applied to calculate adsorption equilibria of methane, ethane and their mixture. At low pressure small pores filled rapidly due to strong wall potentials. The selectivity strongly depended on pressure and pore width. 

In the work of Rowley et al. [1-3], the grand canonical Monte Carlo (gcmc) method was used to simulate Ar interacting with graphite. The surface was approximated as a continuum. In such a case, the sum in Eqn (4.3) is replaced by an integral in the x, y, and z dimensions (the graphite solid) and the potential reduces to a L-J 9-3 form that is a function only of the distance of the atom... 

The first method for simulating chemically reactive systems was proposed by Coker and Watts [11,12]. They presented a modified grand canonical Monte Carlo method wherein the total number of molecules is held fixed but the concentrations of the reacting species is allowed to vary. In their method a molecule is allowed to change species with a probability proportional to the exponential of the difference in chemical potentials between the two components. Thus, their method requires that the chemical potential differences be specified. Coker and Watts applied their method to the reaction... 

Such methods of analysing and describing adsorption data have considerable merit in describing microporosity in porous carbons, which are not crystalline, or for microporous solids of unknown structure, but for zeolites of known structure they add little to our understanding. In such cases, the form of the adsorption isotherms can be modelled by computer simulation using Grand Canonical Monte Carlo methods. In this approach all the parameters are known or can be measured or calculated (see Section 4.5.1) so that the adsorption isotherm can be simulated using a physically well-characterised model. 

As we have aheady said, the grand canonical Monte Carlo provides a mean to determining the chemical potential, and hence, die free energy of the system. In other MC and MD calculations a numerical value for the free energy can always be obtained by means of an integration of thermodynamic relations along a path which links the state of interest to one for which the free energy is already known, for example, the dilute gas or the low-temperature solid. Such a procedure requires considerable computational effort, and it has alow numerical stability. Several methods have been proposed and tested. 

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