BASIC CONCEPTS AND AN OVERVIEW - Foundations of Nanotechnology: Pore Size in Carbon-Based Nano-Adsorbents - page 85

In the future the pore connectivity will be taken into account, so that a more elabo-
rated method will be settled, although so far molecular simulation porosimetry is
not a popular method [1, 2].

1.2.2.7 ADVANCE METHODS BASED ON DENSITY FUNCTIONAL
THEORY
Beside classical methods of pore size analysis, there are many advanced methods.
Some researchers proposed a method based on the mean field theory. Initially this
method was less accurate in the range of small pore sizes, but even so it gave a
more realistic way for evaluation of the pore size distribution than the classical
methods based on the Kelvin equation. More rigorous methods based on molecu-
lar approaches such as grand canonical Monte Carlo (GCMC) simulations and
non-local density functional theory (NLDFT) have been developed and their use
for pore size analysis of active carbons is continuously growing. Let us consider
a one-component fluid confined in a pore of given size and shape, which is itself
located within a well-defined solid structure. We suppose that the pore is open and
the confined fluid is in thermodynamic equilibrium with the same fluid (gas or
liquid) in the bulk state at a given temperature. As the bulk fluid is homogeneous,
its chemical potential is simply determined by the pressure and temperature. The
fluid in the pore is not of constant density and it is subjected to adsorption forces
in the vicinity of the pore walls. This in homogeneous fluid, which is stable un-
der the influence of the external field, is in effect a layer-wise distribution of
the adsorbate. The density distribution can be characterized in terms of a density
profile, p(r), expressed as a function of distance, r, from the wall across the pore.
In the density functional theory (DFT) the statistical mechanical grand canonical
ensemble is used. The appropriate free energy quantity is the grand Helmholtz
free energy, or grand potential functional, D(r). This free energy functional is ex-
pressed in terms of the density profile p(r): then by minimizing the free energy (at
constant, V, T) it is possible in principle to obtain the equilibrium density profile.
For a one-component fluid, which is under the influence of a spatially varying
external potential, the grand potential functional becomes:

( ) ( ) ( ) ( )

∫

W r r = r F r + r Φ −m r r dr

(49)

where F[p(r)] is the intrinsic Helmholtz free energy functional, is the extemal
potential,and the integration is performed over the pore volume V. The F[p(r)]
functional can be separated into an ideal gas term and contributions from the
repulsive and attractive forces between the adsorbed molecules the fluid- fluid
interactions)[1, 2, 11, 36].
Hard-sphere repulsion and pair wise Lenard-Jones potential are usually as-
sumed and a mean field treatment is generally applied to the long-range attraction.
In the earliest local version of density functional theory (LDFT) the Helmholtz
free energy was assumed to be a single valued function of the local density p(r).