Semi-grand potential

The starting point of FVT is the calculation of the semi-grand potential describing the system of Nc colloidal spheres plus depletants as depicted in Fig. 3.6. 

In Fig. 3.10 the semi-grand potential is presented as a function of the colloid volume fraction for given depletant reservoir concentration and size ratio q. Four possible scenarios are considered. Indicated are the common tangent constructions that allow to determine conditions where two (or three) phases coexist. A first criterion for two coexisting binodal composition is equality of the slope because it... 

Fig. 3.10 The dimensionless semi-grand potential 12 as a function of volume fraction (j>. Schematic view of the common tangent construction (straight lines) to determine the phase coexistence in mixtures of colloidal hard spheres and phs. (i) gas-4iquid coexistence, (ii) fluid-solid coexistence, (iii) gas-liquid-solid triple coexistence, and (iv) fluid-solid coexistence near a metastable (dashed lines represent the common tangent construction for this case) gas-liquid coexistence...

We now incorporate the correct depletion thickness into free volume theory presented in Sect. 3.3. We consider the osmotic equilibrium between a polymer solution (reservoir) and the colloid-polymer mixture (system) of interest, see Fig. 4.6. The general expression for the semi-grand potential for Nc hard spheres plus interacting polymers as depletants, see (3.18), is... 

The free volume treatment given for hard spheres + penetrable hard spheres can be extended to the case of asymmetric hard sphere mixtures as follows. The osmotic equilibrium system considered is depicted in a schematic way in Fig. 5.2. We assume the depletion layers are equal to the radii of the small hard spheres. Following the same steps as in Chap. 3 we obtain for the semi-grand potential of the asymmetric hard sphere mixture. 

We now have all the ingredients that make up the semi-grand potential (5.1) of the asymmetric hard sphere mixture. From it we obtain the pressure of the system P and the chemical potential /tj of the large hard spheres using standard thermodynamic relations. 

The starting point for the calculation of the phase behaviour is the semi-grand potential for the system of colloidal rods + phs in osmotic equilibrium with a reservoir with phs, which sets the chemical potential of the phs. This system is depicted in Fig. 6.4. In the free volume approximation (see Sect. 3.3) we can write (3.24)... 

We now have all the contributions to construct the semi-grand potential n(A i, V, T, IJ.2) given in (6.33). In order to obtain the phase behaviour we proceed along the same lines as for the system of pure rods involving the following steps... 

Fig. 6.7 Phase diagrams calculated using free volume theory for spherocylinders with LjD = 20 plus penetrable hard spheres at three size ratios q = 0.3 (left), q = (middle) and q = 2.5 (right). The upper three curves are in the reservoir representation, the lower curves are the system results. The Gaussian form for the ODF was used to minimize the semi-grand potential and compute the coexistence concentrations...

As we noted in Chap. 4, expression (6.33) just like (4.5) only holds for noninteracting depletants. The general expression for the semi-grand potential for hard spherocyhnders plus interacting depletants is... 

Expression (6.46) may be regarded as a generalized free volume theory (GFVT) semi-grand potential for rods plus interacting polymers. Subsequently, we can specify the quantities a and P for interacting polymers. 

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]. 

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]. 

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. )...

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. 

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... 

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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