The semi-grand-canonical technique for polymer blends

2 The semi-grand-canonical technique for polymer blends 

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 

Of course, experiments are done in the canonic ensemble only, and chain identity switches as indicated in Fig. 7.5(b) do not occur, but due to this equivalence between the statistical ensembles that distinction does not matter, in the thermodynamic limit. As will be discussed below, finite size effects are rather different in these various ensembles, and since one is usually able to simulate rather small boxes containing less than 10 chains only, these size effects must be considered carefully. If one is interested in dynamical properties, e.g., the study of initial stages of phase separation (Fig. 7.1), one must use the statistical ensemble that exactly corresponds to the experiment. 

We first discuss the strictly symmetrical situation, where both Na = Nb = N and caa = bb, so the two types of chains differ only by their labels. The order parameter of the unmixing transition can then be written as 

Because of the symmetry of the problem against an interchange of A and B, phase coexistence can occur only for A// = 0, and the critical point must occur for a critical value of the volume fraction 

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