Semigrand ensemble

G[T P] = G = —RTIn A, where A is the isothermal-isobaric partition function and U[T,n ] = —RTInE, where S is the grand canonical ensemble partition function. When a system involves several species, but only one can pass through a membrane to a reservoir, L/(7 jux] = — PTlnT, where T is the semigrand ensemble partition function. The last chapter of the book is on semigrand partition functions. 

If a system contains two types of species, but the membrane is permeable only to species number 1, the natural variables for the system are T, K //, and N2, where N2 is the number of molecules of type 2 in the system. The thermodynamic potential for this system containing two species is represented by U[T, //,]. The corresponding ensemble is referred to as a semigrand ensemble, and the semigrand partition function can be represented by P(71 K /q, N2). The thermodynamic potential of the system is related to the partition function by... 

R. A. Alberty and I. Oppenheim, Use of semigrand ensembles in chemical equilibrium calculations on complex organic systems. J. Chem. Phys. 91, 1824-1828 (1989). 

The advantage of the proposed modification is that is it much more efficient to attempt to increase the size of an existing molecule, than to attempt to place a molecule at a completely random position. In this respect the proposed modification is similar to the semigrand ensemble Monte Carlo simulation technique (2). 

A technique that can be particularly useful for simulation of mixtures is the so-called semigrand ensemble of Kofke and Glandt [93]. A thorough review of this technique and its applications is given in the chapter by D. Kofke (this volume). In that ensemble, ratios of the chemical potential of each component and that of a reference component are specified, whereas the composition of the system is allowed to fluctuate. The equilibrium composition is attained by trial changes of the identity of each species, at constant number of particles (such changes are accepted according to the... 

The semigrand ensemble method can be implemented in different forms for calculation of equilibrium properties, and phase equilibria for inert or reacting mixtures. Recently, it has been applied to simulate phase coexistence for binary polymer blends [85], where advantage was taken of the fact that identity exchanges are employed in lieu of insertions or deletions of full molecules. The semigrand ensemble also provides a convenient framework to treat polydisperse systems (see Section III.F). 

The notion of a semigrand ensemble arises when this decision is made in the context of mixture composition variables. Experience tells us that mole numbers or mole fractions are the most natural way to specify a mixture composition, and indeed anyone unfamiliar with the formalism of thermodynamics may not conceive that there exists an alternative. Of course, the component chemical potentials provide just this alternative, and their selection as independent thermodynamic variables in lieu of mole numbers is no less valid than the substitution of the pressure for the volume. In fact there are very familiar physical manifestations of semigrand ensembles in Nature. For example, in osmotic systems the amount of one component is not fixed, but takes a value to satisfy equality of chemical potential with a solvent bath. Another example is seen in systems undergoing chemical reaction, where the amounts of the various components are subject to chemical equilibrium. 

A marvelous feature of molecular simulation (and thermodynamics in general) is that one need not work with the formalism (ensemble) that is most easily realized physically. Thus one may invoke a semigrand ensemble outside of applications involving osmotic or chemical equilibria. Often this option proves very convenient and desirable. The main point to keep in mind is that one fixes the chemical potential of some components of the mixture and one measures (or calculates) the composition. To proceed, one must surmount the minor conceptual hurdle of working with a system of variable composition absent any plausible reaction mechanism (or even stoichiometry). One can then play many variations on this theme. 

Semigrand ensembles—as the term implies—lie between the canonical and the full grand-canonical ensembles. They are formulated by Legendre transformation of some, rather than all, of the chemical terms. Thus we derive an osmotic semigrand ensemble by the most straightforward variant we transform some terms ( = m), and leave others (j = m + 1,..., c) ... 

We prefer to continue the development using isobaric ensembles, which can be done with semigrand ensembles but not with the fully grand ensemble. We have exercised this option in the Eq. (2.3). Here G = A — PV is the Gibbs free energy and H = U + PV is the enthalpy. 

Our outline for formulating semigrand ensembles with constraints follows naturally from Krishna Pant and Theodorou s [10] development for polymer mixtures. 

Another way to regroup the chemical terms may be suggested by the occurrence of reaction equilibria in the system of interest [11,12]. The resulting formulation is a special case of the constrained semigrand ensemble just described, but it has simplifying features and established notation that make its separate description worthwhile. If the chemical reaction is written in the general form... 

Legendre transformation with respect to the last term yields a reactive semigrand ensemble... 

The fugacity fraction is a convenient quantity because it is bounded between zero and unity, and it has a qualitative correspondence with the mole fraction. The set of fugacity fractions are not independent, as they must sum to unity. Instead the complete set of independent chemical variables is specified by m — 1 fugacity fractions together with the sum Yj=i fj In terms of fugacity fractions the isomolar semigrand ensemble of Eq. (2.5) is expressed... 

It is necessary to know the form of the partition functions to construct transition probabilities that properly sample the ensemble. For instructional purposes we record here the (semiclassical) partition functions that correspond to the osmotic and isomolar semigrand ensembles described above. In both ensembles one must average over moles of the Legendre-transformed species. Thus the osmotic semigrand ensemble partition function is... 

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