Grand-potential functional

Substituting Equations 4.37 and 4.38 into Equation 4.36, the grand potential functional can be written as... 

The value of DFT is evidently dependent on the accessibility and accuracy of the grand potential functional, Si [p(r)]. The usual practice is to treat the molecules as hard spheres and divide the fluid-fluid potential into attractive and repulsive parts. A mean field approximation is used to simplify the former by the elimination of correlation effects. The hard sphere term is further divided into an ideal gas component and an excess component (Lastoskie etal., 1993). The ideal component is considered to be exactly local, since this part of the Helmholtz free energy per molecule depends only on the density at a particular value of r. 

In recent years, a number of investigators have studied the phase equilibria of simple fluids in pores by the application of density functional theory. Semina] studies were carried out by Evans and his co-workers (1985,1986). Their approach was considered to be the simplest realistic model for an inhomogeneous three-dimensional fluid . The starting point was a model intrinsic Helmholtz free energy functional F(p), with a mean-field approximation for the attractive forces and hard-sphere repulsion. As explained in Section 7.6, the equilibrium density profile of the fluid in a pore was obtained by minimizing the grand potential functional. 

For a one-component fluid, which is under the influence of a spatially varying external potential, the grand potential functional becomes... 

The so-called Grand Potential function J = A — G has occasionally been used to characterize thermodynamic states. Show under what conditions this quantity can be related to reversible performance of work and characterize the properties of this function when irreversible processes occur. 

Each individual pore has a fixed geometry, and is open and in contact with bulk gas at a fixed temperature. For this system, the grand canonical ensemble provides the appropriate description of the thermodynamics. In this ensemble, the chemical potential temperature T, and pore volume V are specified. In the presence of a spatially varying external potential Vea, the grand potential functional Qoithe fluid is [11]... 

The equilibrium density profile is determined by minimizing the grand potential functional with respect to the local density. 

The molecular DFT approach [7, 8] places to our disposal the contributions of the free energy, the solid-fluid-interactions and the chemical potential to the grand potential functional on the basis of suitable model conceptions. The final functional expression fl[p] can be differentiated at fixed wall potential v (r,w) and variables of state T,p in order to yield an analytically given relation which enables the calculation of the equilibrated density profile p (z,oj). 

Aside from this, a comparison with the equilibrated mixture results shows that we are now dealing with a situation in which the partial quenching alters the behavior of the chemical potential, and to a much lesser extent the internal energy. As the dipolar density is increased, the HNC equation of the electrolyte breaks down. This is very likely due to a demixing transition, as we could conclude from the stability analysis carried out following the prescriptions of Chen and Forstmann [24, 28], According to them, it is possible to analyze the stability of the grand potential functional for the case of a ion-dipole mixture with equal size particles. The fluctuations in this quantity for the present case can be cast in the form [24],... 

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... 

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... 

Although the essence of DFT is quite simple, the formulation of the functional is chaUenging. For different systems in which the natures of the molecular interaction are distinct, the formulations of the grand potentials can be different. In grand canonical ensemble, the grand potential functional and the intrinsic free energy functional for a fluid system under the external potential fj (R) are related by... 

The minimization of the grand potential functional gives rise to the Euler-Lagrange equation for the LJ system. 

For a mixture of charged and neutral HSs, the grand potential functional takes the similar expression as Eq. (2), and the ideal Helmholtz free energy functional is the summation of the contributions from each species which has the analytic expression as Eq. (12), and the excess Helmholtz free energy functional can be constructed as (Li and Wu, 2004 Yu et al., 2004)... 

Similar to that for simple systems, the grand potential functional Q for rigid molecule system at temperature T, volume V, and chemical potential fi can generally be expressed as... 

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