Grand-potential

Once the mutually consistent volume fraction and segment potential profiles are known, which typically requires an iterative numerical search procedure, one can evaluate thermodynamic quantities. From the above it is clear that the grand potential n = F— is the central quantity of interest. From its definition it is easily seen 

In the case that the pjS are fixed for all components except that of the surfactant (and water), that is when the ionic strength and the free volume bulk concentration are 

Now the excess of surfactant (with respect to the Gibbs plane) is given by  

In the regions far from the central micelle, where the adsorbed amount is not a function of r, it is also possible to compute the limiting interfacial tension y = Mr). This value is used to evaluate the surface pressure at which the 

Coexistence of micelles of different size at certain surfactant chemical potential is always expected. The size distribution can be derived from the SCF calculations. The central quantity is the excess Helmholtz energy of the micelle, F [5,18]. At fixed values of the chemical potentials ( t ), T and p, the excess Helmholtz energy has a minimum at the most likely micelle size. Very close to this minimum, and in first approximation, the excess Helmholtz energy is found to be a quadratic function of the micellar size  

---

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... 

The grand canonical ensemble corresponds to a system whose number of particles and energy can fluctuate, in exchange with its surroundings at specified p VT. The relevant themiodynamic quantity is the grand potential n = A - p A. The configurational distribution is conveniently written... 

To illustrate the complexity of the phase behavior in a more compact way it is instructive to employ a mean-field lattice-gas model. The relative simplicity of the grand potential... 

We begin with the definition of the grand potential f] as a functional of the number density of a fluid [49], p(r)... 

The equihbrium density profile is obtained by minimizing the grand potential, 60,/Sp z) = 0. Henee we obtain... 

The prescription proposed in the original Meister-Kroll-Groot [138,139] theory for hard spheres requires the determination of the local density and the averaged density as two independent variational variables by minimizing the grand potential with respect to these variables. The modification introduced by Rickayzen et al. [143,144] arises from another definition of the average density... 

According to these modifications, the grand potential of hard spheres is the function only of the true density p(r). Similarly to the theory described in the previous subsection, the associative contribution to the free energy... 

Hqq are the average number, the activity of the matrix particles, and the Hamiltonian of the matrix system, respectively. The grand potential of this system is... 

Let us underline some similarities and differences between a field theory (FT) and a density functional theory (DFT). First, note that for either FT or DFT the standard microscopic-level Hamiltonian is not the relevant quantity. The DFT is based on the existence of a unique functional of ionic densities H[p+(F), p (F)] such that the grand potential Q, of the studied system is the minimum value of the functional Q relative to any variation of the densities, and then the trial density distributions for which the minimum is achieved are the average equihbrium distributions. Only some schemes of approximations exist in order to determine Q. In contrast to FT no functional integrations are involved in the calculations. In FT we construct the effective Hamiltonian p f)] which never reduces to a thermo-... 

In order to control the composition of the binary alloy we use the chemical potential pj, and minimize the grand potential Cl. The densities of A and B atoms are obtained from the point distiribution function fj(r) and f2(r) as... 

As usually done in the CVM treatment, we transform the chemical potential terms in the grand potential (given below) il as... 

For a system of N lattice points, the grand potential is given by... 

The grand potential of the binary FCC alloy system is now given by... 

The analog of Eq. (8-211) for the Helmholtz free energy is the grand potential... 

Then from Eq. (8-224) we find the grand potential to be26... 

It can be shown that grand potential of all the atoms in A and B becomes minimum, when A and B have an approximately equal global softness. Extending the idea to the atomic level, when two molecules A and B approach to each other to form a new ... 

Finally, the remaining (/a, Q) representation describing the equilibrium state of an externally open molecular system with the frozen nuclear framework is examined. The relevant partial Legendre transform of the total electronic energy, which replaces N by /a in the list of independent state-parameters, defines the BO grand-potential ... 

The thickness of the layer of the adsorbed molecules is the characteristic distance scale for fractal surface. (3) Van der Waals attraction forces between solid/gas interactions and the liquid/gas surface tension forces are contributed to the grand potential of the system. 

Thermodynamically, we would like to know which material minimizes the free energy of a system containing gaseous 02 and a solid at the specified conditions. A useful way to do this is to define the grand potential associated with each crystal structure. The grand potential for a metal oxide containing NM metal atoms and No oxygen atoms is defined by... 

We can interpret the internal energy in the grand potential as simply the total energy from a DFT calculation for the material. It is then sensible to compare the grand potentials of the different materials by normalizing the DFT energies so that every DFT calculation describes a material with the same total number of metal atoms. If we do this, Eq. (7.1) can be rewritten as... 

The grand potential defined in Eq. (7.2) has one crucial and simple property The material that gives the lowest grand potential is also the material that minimizes the free energy of the combination of gaseous 02 and a solid. In other words, once we can calculate the grand potential as shown in Eq. (7.2), the thermodynamically stable state is simply the state with the lowest grand potential. 

When comparing crystalline solids, the differences in internal energy between different structures are typically much larger than the entropic differences between the structures. This observation suggests that we can treat the entropic contributions in Eq. (7.2) as being approximately constant for all the crystal structures we consider. Making this assumption, the grand potential we aim to calculate is... 

Here is an additive constant that is the same for every material. Because we are only interested in which material has the lowest grand potential, we can set this constant to zero without any loss of generality = 0. If you now com-... 

One feature of Fig. 7.1 is that both of the materials shown on the diagram are thermodynamically favored for some combination of temperature and 02 pressure. This does not have to be the case. Figure 7.2 shows an extension of this grand potential diagram highlighting the thermodynamically favored materials already identified and adding information for a third oxide with... 

Figure 7.1 (a) Schematic grand potential diagram for oxidation of Cu or Ag and (b) the phase... 

Figure 7.2 Extension of the grand potential diagram from Fig. 7.1 with the addition of a metal oxide that is not thermodynamically stable under any conditions.

When we developed expressions for the grand potential above [Eqs. (7.1)—... 

Note that in the standard definition of the grand potential the summation independently extends over the particle numbers of the dilfcrcnt molecular species. For the present problem this implies summation over the numbers M(n) of macro-molecules of length n — 1,2,3,...,... 

---