Semi-grand-canonical simulations

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... 

Fig. 7.5 In dynamic Monte Carlo simuiations of the original Fiory-Huggins modei ehain configurations are relaxed by end-bond rotations, kink jump motions, and 90° crankshaft moves (A). Only such moves are allowed that do not violate the excluded voltmie constraint. In a semi-grand-canonical simulation, where the chemical potential difference A/x between A-and B-monomers is fixed, A-chains are taken out of the system and B-chains are inserted in exactly the same configuration, or vice versa (B). (From Binder. )...

Another technique rests on calculating the structure factor S q). As demonstrated first by Sariban and Binder, though in the framework of a semi-grand-canonical simulation, one can estimate the spinodal curve from a linear extrapolation of versus e/ksT estimating the... 

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. 

Hence simulations can be carried out at fixed values of/only an analog to the semi-grand-canonical technique (where A/j, rather than the composition is fixed) does not exist. As a consequence, it is very important that long-wavelength composition fluctuations relax fast enough. It is this consideration that led Fried et to choose the moves shown in Fig. 7.16 ... 

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%...

The arrows indicate a semi-permeable membrane and the species allowed to permeate is shown within the arrows. The parentheses show a GEMC phase (or region) and the species it contains. The first and the last region are also connected to each other. Using such a scheme, Bryk et al. showed that osmotic Monte Carlo can be successfully used to study the association of two different molecular species when an associating intermolecular potential is included in the simulation. The results agreed well with the more traditional grand-canonical Monte Carlo methods. 

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