Ensemble Gibbs

In contrast to the Gibbs ensemble discussed later in this chapter, a number of simulations are required per coexistence point, but the number can be quite small, especially for vapor-liquid equilibrium calculations away from the critical point. For example, for a one-component system near the triple point, the density of the dense liquid can be obtained from a single NPT simulation at zero pressure. The chemical potential of the liquid, in turn, determines the density of the (near-ideal) vapor phase so that only one simulation is required. The method has been extended to mixtures [12, 13]. Significantly lower statistical uncertainties were obtained in [13] compared to earlier Gibbs ensemble calculations of the same Lennard-Jones binary mixtures, but the NPT + test particle method calculations were based on longer simulations. 

The Gibbs Ensemble MC simulation methodology [17-19] enables direct simulations of phase equilibria in fluids. A schematic diagram of the technique is shown in Fig. 10.1. Let us consider a macroscopic system with two phases coexisting at equilibrium. Gibbs ensemble simulations are performed in two separate microscopic regions, each within periodic boundary conditions (denoted by the dashed lines in Fig. 10.1). The thermodynamic requirements for phase coexistence are that each 

The acceptance criteria for the Gibbs ensemble were originally derived from fluctuation theory [17]. An approximation was implicitly made in the derivation that resulted in a difference in the acceptance criterion for particle transfers proportional to 1/N relative to the exact expressions given subsequently [18]. A full development of the statistical mechanics of the ensemble was given by Smit et al. [19] and Smit and Frenkel [20], which we follow here. A one-component system at constant temperature T, total volume V, and total number of particles N is divided into two regions, with volumes Vj and Vu = V - V, and number of particles Aq and Nu = N - N. The partition function, Q NVt is 

Smit et al. [19] used the partition function given by (10.4) and a free energy minimization procedure to show that, for a system with a first-order phase transition, the two regions in a Gibbs ensemble simulation are expected to reach the correct equilibrium densities. 

The acceptance criteria for the three types of moves can be obtained immediately from (10.4). For a displacement step internal to one of the regions, the probability of acceptance is the same as for conventional constant—iVVT simulations 

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Simulations in the Gibbs ensemble attempt to combine features of Widom s test particle method with the direct simulation of two-phase coexistence in a box. The method of Panagiotopoulos et al [162. 163] uses two fiilly-periodic boxes, I and II. 

The Gibbs ensemble method has been outstandingly successfiil in simulating complex fluids and mixtures. 

For a multicomponent system, it is possible to simulate at constant pressure rather than constant volume, as separation into phases of different compositions is still allowed. The method allows one to study straightforwardly phase equilibria in confined systems such as pores [166]. Configuration-biased MC methods can be used in combination with the Gibbs ensemble. An impressive demonstration of this has been the detennination by Siepmaim et al [167] and Smit et al [168] of liquid-vapour coexistence curves for n-alkane chain molecules as long as 48 atoms. 

Smit B 1993 Computer simulations in the Gibbs ensemble Oomputer Simuiation in Ohemicai Physics vol 397 NATO ASi Series C ed M P Allen and D J Tildesley (Dordrecht Kluwer) pp 173-209... 

Panagiotopoulos A Z 1995 Gibbs ensemble techniques Observation, Prediction and Simuiation of Phase Transitions in Oompiex Fiuids ed M Baus, L F Rull and J-P Ryckaert, vol 460 NATO ASi Series O (Dordrecht Kluwer) pp 463-501... 

Panagiotopoulos A Z, Quirke N, Stapleton M and Tildesley D J 1988 Phase equilibria by simulation in the Gibbs ensemble. Alternative derivation, generalization and applioation to mixture and membrane equilibria Mol. Phys. 63 527-45... 

Panagiotopoulos A Z 1992 Direot determination of fluid phase equilibria by simulation in the Gibbs ensemble a review Mol. SImul. 9 1 -23... 

Panagiotopoulos A Z 1987 Adsorption and oapillary oondensation of fluids in oylindrioal pores by Monte Carlo simulation in the Gibbs ensemble Mol. Phys. 62 701-19... 

Esoobedo F A and de Pablo J J 1996 Expanded grand oanonioal and Gibbs ensemble Monte Carlo simulation of polymers J. Chem. Phys. 105 4391-4... 

Simulating Phase Equilibria by the Gibbs Ensemble Monte Carlo Method... 

Fig. 3.23 The Gibbs ensemble Monte Carlo simulation method uses one box for each of the two plwses. Three types < move are permitted translations within either box volume changes (keeping the total volume constant) and transfer a particle from one box to the other.

Smit B 1993. Computer Simulation in the Gibbs Ensemble. In Allen M P and D J Tildesley (Editor Computer Simulation in Chemical Physics. Dordrecht, Kluwer. NATO ASI Series 397, pp. 173-210... 

Another example of phase transitions in two-dimensional systems with purely repulsive interaction is a system of hard discs (of diameter d) with particles of type A and particles of type B in volume V and interaction potential U U ri2) = oo for < 4,51 and zero otherwise, is the distance of two particles, j l, A, B] are their species and = d B = d, AB = d A- A/2). The total number of particles N = N A- Nb and the total volume V is fixed and thus the average density p = p d = Nd /V. Due to the additional repulsion between A and B type particles one can expect a phase separation into an -rich and a 5-rich fluid phase for large values of A > Ac. In a Gibbs ensemble Monte Carlo (GEMC) [192] simulation a system is simulated in two boxes with periodic boundary conditions, particles can be exchanged between the boxes and the volume of both boxes can... 

In this section we review several studies of phase transitions in adsorbed layers. Phase transitions in adsorbed (2D) fluids and in adsorbed layers of molecules are studied with a combination of path integral Monte Carlo, Gibbs ensemble Monte Carlo (GEMC), and finite size scaling techniques. Phase diagrams of fluids with internal quantum states are analyzed. Adsorbed layers of H2 molecules at a full monolayer coverage in the /3 X /3 structure have a higher transition temperature to the disordered phase compared to the system with the heavier D2 molecules this effect is... 

Another method of simulating chemical reactions is to separate the reaction and particle displacement steps. This kind of algorithm has been considered in Refs. 90, 153-156. In particular. Smith and Triska [153] have initiated a new route to simulate chemical equilibria in bulk systems. Their method, being in fact a generalization of the Gibbs ensemble Monte Carlo technique [157], has also been used to study chemical reactions at solid surfaces [90]. However, due to space limitations of the chapter, we have decided not to present these results. 

The first Monte Carlo study of osmotic pressure was carried out by Panagiotopoulos et al. [16], and a much more detailed study was subsequently carried out using a modified method by Murad et al. [17]. The technique is based on a generalization of the Gibbs-ensemble Monte Carlo (GEMC) method applied to membrane equihbria. The Gibbs ensemble method has been described in detail in many recent reports so we will only summarize the extension of the method to membrane equilibria here [17]. In the case of two phases separated by semi-permeable membranes... 

A. Z. Panagiotopoulos, N. Quirke, M. Stapleton, D. J. Tildesley. Phase equilibria in the Gibbs ensemble. Alternate derivation, generalization and application to mixture and membrane equilibria. Mol Phys 55 527, 1988. 

The simulation of a first-order phase transition, especially one where the two phases have a significant difference in molecular area, can be difficult in the context of a molecular dynamics simulation some of the works already described are examples of this problem. In a molecular dynamics simulation it can be hard to see coexistence of phases, especially when the molecules are fairly complicated so that a relatively small system size is necessary. One approach to this problem, described by Siepmann et al. [369] to model the LE-G transition, is to perform Monte Carlo simulations in the Gibbs ensemble. In this approach, the two phases are simulated in two separate but coupled boxes. One of the possible MC moves is to move a molecule from one box to the other in this manner two coexisting phases may be simulated without an interface. Siepmann et al. used the chain and interface potentials described in the Karaborni et al. works [362-365] for a 15-carbon carboxylic acid (i.e. pen-tadecanoic acid) on water. They found reasonable coexistence conditions from their simulations, implying, among other things, the existence of a stable LE state in the Karaborni model, though the LE phase is substantially denser than that seen experimentally. The re-... 

FIG. 24 Monolayer G-LE coexistence conditions from the simulations of Siepmann et al. (Ref. 369) on a pentadecanoic acid model using Gibbs ensemble Monte Carlo simulation. The filled circles are the simulation results. Experimental results are also shown from Ref. 370 (triangles), Ref. 14 (squares), and Ref. 15 (diamonds). (Reproduced with permission from Ref. 369. Copyright 1994 American Chemical Society.)... 

Medeiros M, Costas ME (1997) Gibbs ensemble Monte Carlo simulation of the properties of water with a fluctuating charges model. J Chem Phys 107(6) 2012-2019... 

Fig. 10.1. Schematic diagram of the Gibbs ensemble MC simulation methodology. Reprinted with permission from [6], 2000 IOP Publishing Ltd...

An interesting extension of the original methodology was proposed by Lopes and Tildesley to allow the study of more than two phases at equilibrium [21], The extension is based on setting up a simulation with as many boxes as the maximum number of phases expected to be present. Kristof and Liszi [22, 23] have proposed an implementation of the Gibbs ensemble in which the total enthalpy, pressure and number of particles in the total system are kept constant. Molecular dynamics versions of the Gibbs ensemble algorithm are also available [24-26]. 

In summary, the Gibbs ensemble MC methodology provides a direct and efficient route to the phase coexistence properties of fluids, for calculations of moderate accuracy. The method has become a standard tool for the simulation community, as evidenced by the large number of applications using the method. Histogram reweighting techniques (Chap. 3) have the potential for higher accuracy, especially if... 

Most methods for the determination of phase equilibria by simulation rely on particle insertions to equilibrate or determine the chemical potentials of the components. Methods that rely on insertions experience severe difficulties for dense or highly structured phases. If a point on the coexistence curve is known (e.g., from Gibbs ensemble simulations), the remarkable method of Kofke [32, 33] enables the calculation of a complete phase diagram from a series of constant-pressure, NPT, simulations that do not involve any transfers of particles. For one-component systems, the method is based on integration of the Clausius-Clapeyron equation over temperature,... 

Gibbs ensemble. Good for obtaining a few points for subcritical phase coexistence between phases of moderate densities does not provide free energies directly. Primarily used to study fluid (disordered) phases. Is a standalone approach, and requires modest programming and computational effort to set up and equilibrate the multiple simulation boxes. Provides accurate coexistence points at intermediate temperatures below the critical point but with sufficient thermal mobility to equilibrate. 

Lopes, J. N. C. Tildesley, D. J., Multiphase equilibria using the Gibbs ensemble Monte Carlo method, Mol. Phys. 1997, 92, 187-196... 

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