Membrane Gas Separation

5.9
The New Model 
Recent work has combined existing transport theories to develop a model through which 
separation performance may be predicted as a function of pore size, shape and composi-
tion, and temperature [23] . As a molecule approaches a pore opening each atom in the 
gas molecule will interact with every atom making up the pore wall through van der Waals 
forces. The difference between the potential energy within the pore E
in
and outside the 
pore E
out
is useful for determining the dominant transport mechanism as well as the neces-
sary driving force – energy barrier component. By assuming that the pore is cylindrically 
shaped the potential energy difference W = E
out
– E
in
can be expressed as
W
E
d
d
cyl
out
=
−
−
⎛
⎝⎜
⎞
⎠⎟
48
42
2
4
12
6
6
επ η
σ
σ
(5.21)
where
ε
ε ε
=
g
s
is the well depth and  
σ
  = (  
σ

g
  +  
σ

s
  )/2 is the kinetic diameter that deter-
mines the forces between the gas molecule g and the pore surface atoms s , which is 
defi ned using the Lorentz – Berthelot mixing rules. The pore size d is defi ned here as the 
distance between the surface nuclei while characterization methods such as PALS use a 
pore size d 
 * 
  which considers the electron cloud thickness  
δ
 d surrounding the surface atoms 
such that, d = d 
 * 
  +  
δ
 d , where  
δ
 d = 3.32 Å [55,56] . Similarly, by assuming a slit - shaped 
pore the potential difference can be expressed as
W
E
d
d
slit
out
=
−
⎛
⎝
⎞
⎠ −
⎛
⎝
⎞
⎠
⎡
⎣
⎢
⎤
⎦
⎥
8
1
5
2
1
2
2
2
10
4
πηεσ
σ
σ
(5.22)
f2.J
f2.J
θ
0,surf
θ
0
θ
0
θ
1
θ
1
θ
0,surf
f1.J
F1.J
f1 + f2 = F2
F1 + F2 = 1
PORE
GAS PHASE
F2.J
Figure 5.9 Schematic model of the total fl ux components through microporous 
membranes [30] . Reprinted from Journal of Membrane Science, 104 , R. S. A. de Lange, K. 
Keizer and A. J. Burggraaf, Analysis and theory of gas transport in microporous sol - gel 
derived ceramic membranes, 81 – 100, Copyright (1995), with permission from Elsevier

---

Modelling Gas Separation in Porous Membranes 
99
The parameters utilized for this approach are from Breck [57] for the gases He, H 
2
, CO 
2
, 
O 
2
, N 
2
and CH 
4
, Poling [58] for the gases CO, Ar, C 
2
H 
6
, n - C 
5
H 
12
and SF 
6
, and the universal 
force fi eld (UFF) values [59] are used for the surface atoms, as summarized in Table 5.2 . 
This potential difference has been termed the suction energy since a positive W translates 
to a suction force of the molecule from the outside to the inside of the pore, while a nega-
tive W translates to a repulsive force directing the molecule away from the pore [23] . 
From this, a new transport mechanism is proposed in [23] as ‘ suction diffusion ’ , where 
enhanced velocities are predicted as the gas molecules are sucked into the pore.
In Figure 5.10 an example of W is demonstrated for a single oxygen molecule entering 
a carbon cylindrical pore from outside where the potential energy is zero, i.e. E
out
= 0. 
This single curve can be used to distinguish the pore size regions for which each transport 
mechanism dominates. Additionally, W can be used to estimate the energy barrier for 
activated diffusion
Δ
E
a
(= | W |) and the energy barrier for surface diffusion
Δ
E
S
(= aq = aW ). 
The critical pore sizes d
min
and d
K
distinguish between the regions where three different 
diffusion mechanisms dominate the transport, namely, activated diffusion ( W
<
0), surface 
diffusion (0
<
W
<
RT ) and Knudsen diffusion ( W
>
RT ).