Application of kinetic Monte Carlo method to equilibrium systems: Vapour–liquid equilibria

doi:10.1016/j.jcis.2011.09.074

Journal of Colloid and Interface Science

Volume 366, Issue 1, 15 January 2012, Pages 216-223

https://doi.org/10.1016/j.jcis.2011.09.074 Get rights and content

Abstract

Kinetic Monte Carlo (kMC) simulations were carried out to describe the vapour–liquid equilibria of argon at various temperatures. This paper aims to demonstrate the potential of the kMC technique in the analysis of equilibrium systems and its advantages over the traditional Monte Carlo method, which is based on the Metropolis algorithm. The key feature of the kMC is the absence of discarded trial moves of molecules, which ensures larger number of configurations that are collected for time averaging. Consequently, the kMC technique results in significantly fewer errors for the same number of Monte Carlo steps, especially when the fluid is rarefied. An additional advantage of the kMC is that the relative displacement probability of molecules is significantly larger in rarefied regions, which results in a more efficient sampling. This provides a more reliable determination of the vapour phase pressure and density in case of non-uniform density distributions, such as the vapour–liquid interface or a fluid adsorbed on an open surface. We performed kMC simulations in a canonical ensemble, with a liquid slab in the middle of the simulation box to model two vapour–liquid interfaces. A number of thermodynamic properties such as the pressure, density, heat of evaporation and the surface tension were reliably determined as time averages.

Graphical abstract

Highlights

► Novel application of Kinetic Monte Carlo (kMC) to describe equilibrium systems. ► An advantage of the kMC is the absence of discarded trial moves of molecules. ► kMC is more effective than traditional MC for analysis of inhomogeneous systems. ► The chemical potential is determined directly within the framework of kMC.

Introduction

Monte Carlo (MC) simulations with the Metropolis algorithm [1], [2] have been widely used in studying equilibrium properties of physical systems, for example [3], [4], [5], [6], [7], [8], [9]. In order to analyse systems dynamically, the kinetic Monte Carlo (kMC) has been developed over the past few decades [10], [11], [12], [13], [14]. To date, it has been mainly applied to crystal growth, atomic diffusion and chemical reactions, where time-dependent functions are required. There are no obstacles for using kMC in studying equilibrium systems for which the principal tool of simulation is the Monte Carlo scheme with Metropolis algorithm. The difference between two approaches lies in the way of averaging of thermodynamic functions. The latter is based on ensemble averages, while the kMC uses time averaging, which is equivalent by definition [15]. However, from a technical viewpoint, the kMC has a clear distinction in that no trial moves are discarded; on the other hand, many attempts are rejected in the conventional Monte Carlo when the density is very high, either in the bulk fluid or in the adsorption system with strong inhomogeneity in the fluid density. For such cases, the acceptance ratio is very small, leading to very small maximum displacement length, which results in significant correlation of consecutive states and, consequently, slow exploring of the phase space [1]. Another problem of the standard MC method when dealing with the case of inhomogeneous system is that a molecule to be moved or deleted is chosen randomly with no consideration of its energy. This leads to relative lack of statistical data accumulated in the rarefied region of the non-uniform system, for example, in case of vapour–liquid equilibrium. The kMC does not have such a drawback, and therefore, it allows us to reliably determine the pressure, density and other thermodynamic properties of the vapour phase coexisting with the liquid phase. The aim of the present work is to formulate a simulation scheme in the framework of kMC for highly inhomogeneous systems, such as the gas–solid interface using argon vapour–liquid equilibrium as an example. To this end, we use a canonical ensemble with a liquid slab in the middle of the simulation box, surrounded by the gas phase at two ends. Effects of temperature on the saturation pressure, the densities of coexisting phases, the molar heat of evaporation and the surface tension are presented to illustrate the potential of the kMC scheme. This is the first time in the literature that the kMC is applied to solve equilibrium problems of fluid phase equilibria.

Section snippets

Model

The details of the kinetic Monte Carlo method are described elsewhere [10], [11], [12], [13], [14] in terms of rates of elementary processes (for example, spin flipping). The rate ν_i of a process i is the inverse of the so-called waiting time τ_i, which depends on the current state of i. The total rate R of the system composing of many processes is the sum of the rates of these processes: $R = \sum_{i = 1}^{N} ν_{i}$ where N is the number of processes in the system (volume) under consideration.

On average, one event

Simulation details

We have performed simulations for a homogeneous fluid and an inhomogeneous fluid. The homogeneous fluid (by the example of argon at its boiling point) was studied to confirm the workability of the kMC scheme and to compare the simulation results with those obtained with the standard Metropolis algorithm. In the latter case, we modelled the vapour–liquid interface in order to evaluate the parameters corresponding to the vapour–liquid equilibrium at various temperatures, i.e. the saturation

Application of the kMC to model of a homogeneous fluid

In this section, we present results of simulation of the homogeneous argon at its boiling point (87.29 K) in the canonical ensemble with the kMC and with the standard Metropolis algorithm. In the latter case, the maximum displacement length, r_m, was adjusted to provide the acceptance ratio of 0.25σ during the equilibration stage. In both cases, we used the same number of MC steps for equilibration and averaging, namely, 2 × 10⁶ and 5 × 10⁶ MC steps, respectively.

Fig. 1 presents the pressure of argon

Conclusion

An approach based on the kinetic Monte Carlo method and the time averaging procedure is developed for equilibrium systems. The approach was shown to be an effective tool for the analysis of vapour–liquid equilibrium in canonical ensemble and prediction of thermodynamic properties, the saturation pressure, the densities of coexisting phases, the heat of evaporation and the surface tension over a wide temperature range. An advantage of the approach is the absence of discarded Monte Carlo trial

Acknowledgment

This work is supported by Russian Foundation for Basic Research (Project No 11-03-00129-a). Support from the Australian Research Council is also acknowledged.

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Application of kinetic Monte Carlo method to equilibrium systems: Vapour–liquid equilibria

Abstract

Graphical abstract

Highlights

Introduction

Section snippets

Model

Simulation details

Application of the kMC to model of a homogeneous fluid

Conclusion

Acknowledgment

Fluid Phase Equilib.

Fluid Phase Equilib.

Fluid Phase Equilib.

J. Comput. Phys.

Comput. Methods Appl. Mech. Eng.

J. Comput. Phys.

Physica

Computer Simulation of Liquids

Understanding Molecular Simulation

J. Chem. Phys.

J. Chem. Phys.

J. Phys. Chem. B

Fluid Phase Equilib.

J. Phys. Chem.

Phys. Rev. B