Page 39 - Adsorption by Powders and Porous Solids

24 ADSORPTION BY POWDERS AND POROUS SOLJDS

containing a mixture of components, which is characterized by the grand canonical
variables T, V, p,, . . ., pi, we may write
S2 = F-ZpiNi (1.10)

where F is the Helmholtz free energy (see Chapter 2) and Ni is the number of mole-
cules of component i. For a single-component fluid in the presence of a spatially
varying external potential V,,(r), the grand potential functional can be expressed in
the form
Qb(r)l = Fb(r)l- dr p(r) [P - V,,(r)l, (1.11)
where p is the local fluid density at position r and the integration is performed over
the pore volume.
The Helmholtz energy F represents the intrinsic free energy in the absence of any
external field, whereas 52 is dependent on all the interactions within the pore together
with a surface contribution. When Sa is allowed to vary in response to a change in
p(r), its overall minimum corresponds to the equilibrium density profile of the
system. The equilibrium density profile is therefore determined by minimizing the
grand potential functional with respect to p(r). Since p(r) is the local density, the
amount adsorbed (usually expressed as the surface excess number of molecules
adsorbed) must be obtained by integration over the internal volume of the pore. By
repeating this procedure for different values of p (and hence values of p/pO) it is
possible to construct the adsorption isotherm.
The value of DFT is evidently dependent on the accessibility and accuracy of the
grand potential functional, S2 b(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 corre-
lation effects. The hard sphere term is further divided into an ideal gas component
and an excess component (Lastoskie et al., 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.
The evaluation of the excess free energy is a more difficult problem. This is
because in the inhomogeneous fluid the energy distribution is non-local, that is it
depends on the correlations within the overall density profile. Various attempts have
been made to overcome this difficulty by the introduction of weighting or smoothing
functions (Gubbins, 1997). This approach has led to the development of the non-local
density functional theory (NLDIT), which inter alia has been used for the derivation
of the pore size distribution from adsorption isotherm data (see Chapter 7). The use
of DFT and MC simulation for the study of micropore filling is also under active
investigation (see Chapter 8).
With a number of fairly simple systems, excellent agreement has been obtained
between the corresponding Dm-predicted and MC-generated isotherms, 2-D phase
transitions and adsorption energies. These are encouraging results, but it must be kept
in mind that the computational procedures are not entirely independent. As we have
seen, they are dependent on the same model parameters of adsorbent structure and
potential functions. At present, there are only a few porous adsorbents which have the