BASIC CONCEPTS AND AN OVERVIEW - Foundations of Nanotechnology: Pore Size in Carbon-Based Nano-Adsorbents - page 94

assumed pore geometry. For the wetting films formed on a concave surface of
spherical pores, the following relationship is valid:

( )

2 g r

� h � � h � �� �� RT p

( )

m

Π =Π h exp − +Π exp − + =− ln ln

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1 2

l l rh − v p

ƒ „ ƒ „ ƒ„ ƒ„ (70)

1 2 m 0

As demonstrated previously, the surface tension of a liquid adsorbate depends on
the meniscus radii , which seems to be particularly important for the pores at the
borderline between micropores and mesopores. Similar to the work completed by
Miyahara and co-workers, the GTKB was used in this work. The GTKB equation
for the cylindrical interface (capillary condensation) can be written as follows:

( )

g r

d
=−

m

1

g r (71)

∞ m

In Eq. (71), denotes the surface tension of the bulk fluid, and is the displacement
of the surface at zero mass density relative to the tension surface. The physical
meaning of and its impact on the spinodal condensation point was presented
previously. The stability condition of the wetting film was formulated earlier by
Derjaguin et al. as. Obviously, Capillary Condensation/Evaporation in Spherical
Cavities both the critical film thickness, , and the critical capillary radius, , cor-
responding to the film collapse, are determined from 42

d Π (h) |

= 0

(72)

hh rr == ,

dh

cr cr

The condition given by Eq. (72) determines the spontaneous spinodal conden-
sation when the adsorbed film thickness becomes mechanically unstable.
A combination of Eqs. (69)-(72) gives the relation between both the critical
film thickness and the critical capillary radius as a function of the relative pressure
for the spherical pore geometry. The solution of this system of algebraic equations
can be obtained by chord or other standard numerical procedures. The thickness
of the adsorbed film in equilibrium with the meniscus for the spherical pore is
given by:

d
g −

� �

2 v 1

€ ‚

m ∞

rh −

ƒ „

( ) ( )

RT ln p / p =Π h v +

e

(73) 0 e m

rh −

e

For constant surface tension, Eqs. (72) and (73) reduce to the classical DBD equa-

tions, which are described in the series of papers published by scientits. For ap-
plication purposes, we can derive the analytical formulas for the calculation of
the equilibrium transition in the considered spherical pore geometry according
this equation. As we mentioned above, for the spherical pore, the system of equa-