Grand canonical ensemble method

Indirect Methods Test particle method Grand canonical ensemble method Biased sampling methods Thermodynamic integration... 

The main idea of a lattice model is to assume that atomic or molecular entities constituting the system occupy well-defined lattice sites in space. This method is sometimes employed in simulations with the grand canonical ensemble for the simulation of surface electrochemical proceses. The Hamiltonians H of the lattice gas for one and two adsorbed species from which the ttansition probabilities 11 can be calculated have been discussed by Brown et al. (1999). We discuss in some detail MC lattice model simulations applied to the electrochemical double layer and electrochemical formation and growth two-dimensional phases not addressed in the latter review. MC lattice models have also been applied recently to the study the electrox-idation of CO on metals and alloys (Koper et al., 1999), but for reasons of space we do not discuss this topic here. 

Chapters 10 and 11 cover methods that apply to systems different from those discussed so far. First, the techniques for calculating chemical potentials in the grand canonical ensemble are discussed. Even though much of this chapter is focused on phase equilibria, the reader will discover that most of the methodology introduced in Chap. 3 can be easily adapted to these systems. Next, we will provide a brief presentation of the methods devised for calculating free energies in quantum systems. Again, it will be shown that many techniques described previously for classical systems, such as PDT, FEP and TI, can be profitably applied when quantum effects are taken into account explicitly. 

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). 

There are many excellent reviews on the standard molecular dynamics method dealing with calculations in the microcanonical ensemble as well as on the Monte Carlo method involving calculations in the canonical, isothermal isobaric, and grand canonical ensemble (< ). In the present article, we shall limit ourselves exclusively to those developments that have taken place since the work of Andersen (4). In the molecular dynamics method, the developments are the constant-pressure, constant-temperature, constant-temperature-constant-pressure, variable shape simulation cell MD, and isostress calculations in the Monte Carlo method, it is the variable shape simulation cell calculation. 

Wongkoblap et al.307 study Lennard-Jones fluids in finite pores, and compare their results with Grand canonical ensemble simulations of infinite pores. Slit pores of 3 finite layers of hexagonally arranged carbon atoms were constructed. They compare the efficiency of Gibbs ensemble simulations (where only the pore is modelled) with Canonical ensemble simulations where the pore is situated in a cubic cell with the bulk fluid, and find that while the results are mostly the same, the Gibbs ensemble method is more efficient. However, the meniscus is only able to be modelled in the canonical ensemble. 

This review discusses a newly proposed class of tempering Monte Carlo methods and their application to the study of complex fluids. The methods are based on a combination of the expanded grand canonical ensemble formalism (or simple tempering) and the multidimensional parallel tempering technique. We first introduce the method in the framework of a general ensemble. We then discuss a few implementations for specific systems, including primitive models of electrolytes, vapor-liquid and liquid-liquid phase behavior for homopolymers, copolymers, and blends of flexible and semiflexible... 

Fig. 8. End-to-end vector autocorrelation function for polymer chains obtained by different simulation methods (1) short dashed line, canonical ensemble (2) dash-dot-dotted line, grand canonical ensemble (3) dashed line, multidimensional parallel tempering (4) dash-dotted line, expanded grand canonical ensemble (5) solid line, hyperparallel tempering.

In the DFT method [80], each individual pore has a fixed geometry and is in open contact with the bulk vapor-phase adsorbate at a fixed temperature. For this system, the grand canonical ensemble provides the appropriate description of the thermodynamics. In this ensemble, the chemical potential jx, temperature T, and pore volume F are specified. In the presence of a spatially varying externd potential Fext, the grand potential functional of the fluid is [15]... 

Once the ideal heterogeneous solids are prepared, the adsorption process is simulated through a continuum space Monte Carlo method in the grand canonical ensemble [26, 27]. 

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