Grand canonical free energy

When applied to the grand canonical free energy functional (47) the above condition leads to the following integral equation... 

AB diblock copolymers in the presence of a selective surface can form an adsorbed layer, which is a planar form of aggregation or self-assembly. This is very useful in the manipulation of the surface properties of solid surfaces, especially those that are employed in liquid media. Several situations have been studied both theoretically and experimentally, among them the case of a selective surface but a nonselective solvent [75] which results in swelling of both the anchor and the buoy layers. However, we concentrate on the situation most closely related to the micelle conditions just discussed, namely, adsorption from a selective solvent. Our theoretical discussion is adapted and abbreviated from that of Marques et al. [76], who considered many features not discussed here. They began their analysis from the grand canonical free energy of a block copolymer layer in equilibrium with a reservoir containing soluble block copolymer at chemical potential peK. They also considered the possible effects of micellization in solution on the adsorption process [61]. We assume in this presentation that the anchor layer is in a solvent-free, melt state above Tg. The anchor layer is assumed to be thin and smooth, with a sharp interface between it and the solvent swollen buoy layer. 

The fraction / of charged monomers on an annealed polyelectrolyte is measured in a titration experiment. The simplest description of this experiment [90] is to assume that the polymer is homogeneous and to minimize its grand canonical free energy (its free energy at constant pH). In a mean field approach, the free energy of one chain is... 

Our lattice model is schematically depicted in Figure 2. In its most general form, the grand canonical free energy per unit area,/, is written as... 

One can prove that in thermal equilibrium, in a grand canonical ensemble (i.e., volume, chemical potential, and temperature are fixed), the grand canonical free energy Q(p(r)) of a system can be written as a functional of the one-body density p(r) alone, which will depend on the position r in inhomogeneous systems. The density distribution peq(r) which minimizes the grand potential functional is the equilibrium density distribution. This statement is the basis of the equilibrium density functional theory (DFT) for classical fluids which has been used with great... 

It remains to find F = F [c]. To this end, one can introduce an external field h T(i>(r)ll acting on each half-bead. The grand canonical free energy, dimer system in the presence of the field is defined by... 

Finally, we note that the grand canonical free energy for an ideal system of dimers is... 

A function of appropriately chosen independent variables, from which all thermodynamic properties are derivable by differentiations alone, with no integrations required, is said to be a thermodynamic potential. Examples are t/(S, V, n), S(U, V, n), F(T, V, n), G(T, p, n), and the grand-canonical free energy -pV=fl(T, V, s). Intensive functions of intensive arguments, from which all the intensive properties of the system are derivable, are also called thermodynamic potentials examples are p)f n( p)> l (T,p), and -p(T, i). These are not essentially different from the corresponding extensive potentials, here U, S, F, and H, respectively. 

The stability of the micelle is governed by the grand canonical free energy... 

The osmotic pressure is obtained by minimization of the total grand canonical free energy GV with respect to the volume v ... 

Diblock copolymers are known to form mesophases[9,10] in selective solvents micelles, lamellae, worm-like micelles. .. These aggregation effects are relevant for the interfacial behavior of the copolymers the bulk solution acts as a reservoir for the adsorbed layer and imposes the chemical potential //ex- To study the adsorbed layer we write the siuface grand canonical free energy of the layer as... 

In the density functional theoiy (DFT) the statistical mechanical grand canonical ensemble is utilized. The appropriate free energy quantity is the grand Helmholtz free energy, or grand potential functional, 2(r. This free energy functional is expressed in terms of the density... 

In a canonical ensemble, the system is held at fixed (V, T, N). In a grand canonical ensemble the (V, T p) of the system are fixed. The change from to p as an independent variable is made by a Legendre transfomiation in which the dependent variable, the Flelmlioltz free energy, is replaced by the grand potential... 

In the canonical example, we could estimate the free energy difference between two runs by examining the overlap in their probability distributions. Similarly, in the grand canonical ensemble, we can estimate the pressure difference between the two runs. If the conditions for run I arc f//1. V. > ) and for run 2 (po, VjK), then... 

Chapters 10 and 11 cover methods that apply to systems different from those discussed so far. First, the techniques for calculating chemical potentials in the grand canonical ensemble are discussed. Even though much of this chapter is focused on phase equilibria, the reader will discover that most of the methodology introduced in Chap. 3 can be easily adapted to these systems. Next, we will provide a brief presentation of the methods devised for calculating free energies in quantum systems. Again, it will be shown that many techniques described previously for classical systems, such as PDT, FEP and TI, can be profitably applied when quantum effects are taken into account explicitly. 

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