chapter 4 section 4.2

SEMIGRAND ENSEMBLE

  The most straightforward approximation to a continuous mixture is based in the canonical ensemble and thus takes a finite sample from a distribution of diameters . However, this approach is sensitive to finite size effects and, moreover, it is not practical when phase equilibria are considered as it is difficult to ensure chemical potential equilibration of each component. A better choice is a grand canonical simulation in which particles of different species are inserted and removed according to the configurational energy and the imposed chemical potential of that component. In this way a truly continuous distribution can be realized and phase equilibrium can be more easily treated. However, at the high densities of the liquid-solid equilibrium the insertion probability in both phases is too low to obtain reasonable statistics; further, the need to maintain the crystal structure in the solid makes insertions especially problematic.
The semigrand ensemble provides an alternative representation that combines the best features of the canonical and grand canonical ensembles for the study of continuous mixtures [85,86]. A simulation in this ensemble has the total number of particles fixed, but the species identity of each particle is allowed to change, giving rise to a truly continuous distribution. Although the chemical potentials are imposed in a way similar to the grand-canonical ensemble, insertion of particles is avoided, so the method is suitable for high densities and crystalline phases.
We consider a system of N hard spheres with diameters distributed according to . The isobaric semigrand canonical free energy Y is defined by a Legendre transform of the Gibbs free energy G. In the polydisperse limit,

or, in differential form,

Here, is the chemical potential as a function of , and is the diameter of an arbitrarily chosen reference component. Also, H is the enthalpy, is the reciprocal temperature, P is the pressure and V the volume of the system. The isobaric semigrand canonical potential Y is a function of the independent variables T,P and N and it is a functional of the chemical potential difference function . In a simulation these independent variables must be fixed while the thermodynamic conjugates and are allowed to fluctuate. This implies that the composition can be known only after the simulation has been performed. Because the total number of particles is fixed the chemical potential of the reference has still to be computed. Once it is determined, the entire chemical potential distribution is known. The method is therefore well suited for phase equilibrium in continuous mixtures: for a given temperature and distribution in chemical potential differences one needs match only the values of the pressure and the reference chemical potential in both phases. This is far simpler than matching the entire distribution in the canonical way.
We are interested in determining the influence of polydispersity on the hard sphere fluid-solid transition. Although the composition distribution and hence the polydispersity cannot be imposed directly, it can be expected that its form will be much like that of the imposed activity-ratio distribution [86]. Therefore we choose the following quadratic form for the chemical potential difference function

which gives rise to a Gaussian activity distribution that peaks at , with width . In the limit , the pure monodisperse phase is recovered. For small the mixture is ideal and the composition will be Gaussian with the peak near .
The choice of eqn. 4.3 converts Y from a functional of to a function of and . The fundamental thermodynamic equation now reads

where and are the first and second moment of the composition about . The nth such moment is defined as

In a semigrand Monte Carlo simulation, particles sample diameters in addition to the usual sampling of positions within the simulation box. Diameters are sampled by selecting a particle at random, changing its diameter by a small amount, and accepting with probability in accord with the Metropolis algorithm. Details may be found in [87].

chapter 4 section 4.2