Grand-canonical simulations

In a dense system, the acceptance rate of particle creation and deletion moves will decrease, and the number of attempts must be correspondingly increased eventually, there will come a point at which grand canonical simulations are not practicable, without some tricks to enliance the sampling. 

In Ihc canonical, microcanonical and isothermal-isobaric ensembles the number of particles is constant but in a grand canonical simulation the composition can change (i.e. the number of particles can increase or decrease). The equilibrium states of each of these ensembles are cha racterised as follows ... 

In the studies by Skipper et al. the number of water layers (and thus molecules) was fixed on the basis of experimental evidence consequently, the stable states or degrees of swelhng were presumed. Quite differently, Karaborni et al. [44] determined, by means of a combination of GCMC and MD, the number of water molecules directly from a series of simulations in which the distance between montmorillonite planes was varied systematically. They observed that swelling proceeded from the dry state through the formation of one, three, and then five layers of water. This is very different from the usually beheved hydration sequence from one layer to two, then to three layers, and so on, which has been intrinsically assumed by Skipper and coworkers. The authors conclude that the complex swelling behavior accounts for many of the experimental facts. This work demonstrates impressively the power of the grand canonical simulation method. 

Table HI compiles MC results obtained over the years for the critical temperature and critical density of the RPM. Table in includes also results from the cluster calculations of Pitzer and Schreiber [141]. In a critical assessment of earlier work [40, 141, 179-181, 246], Fisher deduced in 1994 that T = 0.052-0.056 and p = 0.023-0.035 represent the best values [15]. Since then, however, the situation has substantially changed. Caillol et al. [53,247] performed simulations of ions on the surface of a four-dimensional hypersphere and applied finite-size corrections. Valleau [248] used his thermodynamic-scaling MC for systems with varying particle numbers to extract the infinite-size critical parameters. Orkoulas and Panagiotopoulos [52] performed grand canonical simulations in conjunction with a histogram technique. All studies indicate an insufficient treatment of finite-size effects in earlier work. While their results do not agree perfectly, they are sufficiently close to estimate T = 0.048-0.05 and p = 0.07-0.08, as already quoted in Eq. (6). Critical points of some real Coulombic systems match quite well to these figures [5]. The coexistence curve derived by Orkoulas and Panagiotopoulos [52] is displayed in Fig. 9.

The distributions of the Xe atoms among the cages of the zeolite are provided by the relative intensities of the Xe1 Xe2, Xe3,. ..Xe8 peaks seen individually in the 129Xe NMR spectrum. Thus, the NMR experiment provides a direct measure of the distribution of Xe atoms among the cavities, e.g., what fraction of the zeolite cages have 5 exactly Xe atoms This is reproduced very well by the grand canonical simulations described above. 

Figure 5 Density relaxations in Monte Carlo simulations of the geometry shown in Fig. 4 with conditions same as in Fig. 3 /3fi = -5.5) (a) Grand canonical simulations. (6) Simulation with mass conservation. The solid line, dotted line, and the open circles are the Kawasaki dynamics, ideal diffusion, and the grand canonical result shown in (a) rescaled by td with ro = 2 gmcs. The inset shows the initiail diffusion-limited regime in the logarithmic scale.

Fig. 6. Grand-canonical simulation of a Lennard-Jones monomer system. Particles interact via a shifted and truncated Lennard-Jones potential of the form Elj =...

Fig. 13. Illustration of the grand-canonical simulation technique for temperature UbT/e = 1.68 and p = pcoex- A cuboidal system geometry 13.8cr x 13.8cr x 27.6cr is used with periodic boundary conditions in all three directions. The solid line corresponds to the negative logarithm of the probability distribution, P p), in the grand canonical ensemble. The two minima correspond to the coexisting phases and the arrows on the p axis mark their densities. The height of the plateau yields an accurate estimate for the interfacial tension, yLV- The dashed line is a parabohc fit in the vicinity of the liquid phase employed to determine the compressibihty. Representative system configurations are sketched schematically. From [62]...

M. Muller (1999) Miscibility behavior and single chain properties in polymer blends a bond fluctuation model study. Macromol. Theory Simul. 8, pp. 343-374 M. Muller and K. Binder (1995) Computer-simulation of asymmetric polymer mixtures. Macrrmolecules 28, pp. 1825-1834 ibid. (1994) An algorithm for the semi-grand-canonical simulation of asymmetric polymer mixtures. Computer Phys. Comm. 84, pp. 173-185... 

Grand-canonical simulations are somewhat more cumbersome to use than canonical or isothermal-isobaric ensemble simulations. Their implementation often requires exploratory work, because it is generally easier to anticipate or specify the density, pressure or composition of a system (one usually has some reference or an intuitive feeling about such properties), rather than its chemical potentials. 

In the grand canonical ensemble the conserved properties are the chemical potential, the volume and the temperature. It can sometimes be more convenient to perform a grand canonical simulation at constant activity, z, which is related to the chemical potential p by. 

Grand Canonical simulations have been used fairly successfully in simulating singlecomponent systems. More recent papers show that the method can also be used to simulate binary systems and also mixturesl . 

We therefore expect to have to look for phase transitions in two coupled order parameters, density and orientation. To vary the density in the simulation we will employ a grand-canonical simulation technique using the configurational bias scheme for chain insertion and deletion... 

Configurational-bias methods trace their ancestry to biased sampling for lattice polymer configurations proposed by Rosenbluth and Rosenbluth [85]. Development of configurational-bias methods for canonical and grand canonical simulations and for continuous-space models took place in the early 1990s [86-90] and dramatically expanded the range of intermolecular potential models that can be studied by the methods described in the previous sections. 

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