Semi-grand ensemble

Another triek is applieable to, say, a two-eomponent mixture, in whieh one of the speeies. A, is smaller than the other, B. From figure B3.3.8 for hard spheres, we ean see that A need not be particularly small in order for the test partiele insertion probability to elimb to aeeeptable levels, even when insertion of B would almost always fail. In these eireumstanees, the ehemieal potential of A may be detemiined direetly, while that of B is evaluated indireetly, relative to that of A. The related semi-grand ensemble has been diseussed in some detail by Kofke and Glandt [110]. 

At coexistence, the chemical potentials of all species i in the two phases, are equal p = pf. In the notation of the semi-grand ensemble, this means that, at coexistence, the temperature and pressure of the two phases are equal, as are all Api, and finally also the chemical potential p i of the reference compound. Now consider what happens if we supersaturate the parent phase, for instance by compression (the analysis for the case of supercooling follows by analogy). In the semi-grand ensemble we perform this supersaturation by increasing P, while keeping T and all Ap constant. Note that this route need not correspond to the physical route for supersaturation. The reason is the physical route is (usually) to supersaturate at constant composition. But in that case, all Ap, change by different amounts, and this is precisely the factor that complicates the analysis of nucleation in multicomponent systems. 

Note that, as we are working in the semi-grand ensemble where we keep all Ap, constant, we have thus achieved equality of the chemical potentials in the two phases... 

Computationally, polydispersity is best handled within a grand canonical (GCE) or semi-grand canonical ensemble in which the density distribution p(a) is controlled by a conjugate chemical potential distribution p(cr). Use of such an ensemble is attractive because it allows p(a) to fluctuate as a whole, thereby sampling many different realizations of the disorder and hence reducing finite-size effects. Within such a framework, the case of variable polydispersity is considerably easier to tackle than fixed polydispersity The phase behavior is simply obtained as a function of the width of the prescribed p(cr) distribution. Perhaps for this reason, most simulation studies of phase behavior in polydisperse systems have focused on the variable case [90, 101-103]. 

For symmetrical polymer blends (as well as weakly asymmetrical ones) the problem of hydrodynamical slowing down of long wavelength concentration fluctuations can be elegantly avoided by carrying out the simulation in the semi-grand-canonical ensemble rather than the canonical ensemble only the total number of chains n = is fixed, while the ratio... 

The number of chains n in a simulation in the semi-grand-canonical ensemble is constant only the order parameter m (eq. [7.4]) can fluctuate. Therefore the first factor on the right hand side of eq. (7.8) is constant and cancels out from the Monte Carlo averages. 

Cifra et a/. use a wall-monomer interaction = wb but consider different volume fractions (f>A < >b the bulk of their thin film system, carrying out the simulation in the canonical ensemble, while the data of Fig. 7.21(b) are obtained from semi-grand-canonical techniques. The simulation by Cifra et a/. is of great interest as a first step towards the modeling of the situation used in experiment, where one normally has two inequivalent interfaces one is the interface between the thin film of the polymer blend and an adsorbing substrate, and the other is an interface between the film and vacuum or air, respectively. Cifra et a/. achieve this... 

We performed Monte Carlo in the constant-pressure, semi-grand-canonical ensemble of the type described in [52]. In such a simulation it is not possible to impose the size distribution of the particles directly, but the size distribution can be controlled through variation of the imposed activity-ratio distribution function. In our simulations we imposed a Gaussian activity distribution and a typical particle size distribution function is shown in Fig. 14. 

Fig. 14. Typical particle size distribution functions from Monte Carlo simulations in the constant-pressure, semi-grand-canonical ensemble in a bulk liquid and solid. At that pressure the volume fraction of the liquid is = 0.5775 at a polydispersity of 10%. The volume fraction of the solid in coexistence with the liquid is = 0.6196 and has a polydispersity of 4.2%...

We stress that, for every component, the chemical potentials in the parent phase and in the critical nucleus are the same. In the absence of the Laplace pressure, the chemical potentials in phase a would be lower than those in phase yS. The effect of the Laplace pressure is to compensate this difference for every component i. At first sight, it would seem that the computation of Ap is an intractable problem for a multicomponent system - to satisfy the condition that pf = pf for all i, it is not enough to compress phase a we should also change its composition. The problem is greatly simplified if we make use of the semi-grand canonical ensemble. In the semigrand ensemble, the independent variables that describe the state of an -component system are the temperature T, the pressure P, the total number of particles N and the set of n - 1 differences in the chemical potential (Ap,) between a reference species (say, species 1) and all other species i 1. The number of components n can be infinite. 

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