Membrane Gas Separation

5.3
Surface Diffusion 
Surface diffusion is the diffusion mechanism which dominates in the pore size region 
between activation diffusion and Knudsen diffusion [29,30,34,35] . A model that well 
described the surface diffusion on the pore walls was proposed many years ago [36] . It 
was shown to be consistent with gas and vapour transport parameters in porous polymeric 
membranes (nucleopore). When the pore size decreased below a certain level, which 
depends on both membrane material and the permeating gas, the gas permeability coef-
fi cient exceeds the value for free molecular fl ow (Knudsen diffusion), especially in the 
case of organic vapours. Note that surface diffusion usually occurs simultaneously with 
Knudsen diffusion but it is the dominant mechanism within a certain pore size region 
discussed later in this chapter. Since surface diffusion is also a form of activated diffusion, 
the energy barrier
Δ
E
S
is the energy required for the molecule to jump from one adsorp-
tion site to another across the surface of the pore. By allowing the energy barrier to be 
Table 5.1  Transport mechanisms 
Mechanism
Schematic
Process
Activated diffusion
constriction energy barrier 
Δ
E
a
Surface diffusion
adsorption - site energy barrier 
Δ
E
s
Knudsen diffusion
Direction and velocity 
d
u

---

Modelling Gas Separation in Porous Membranes 
89
proportionate to the enthalpy of adsorption, Gilliland et al. [35] established an equation 
for the surface diffusion coeffi cient expressed here as
D
D
aq
RT
S
S
=
−
⎛
⎝
⎞
⎠
* exp
(5.9)
where
D
S
*
is a pre - exponential factor depending on the frequency of vibration of the 
adsorbed molecule normal to the surface and the distance from one adsorption site to the 
next. The quantity q ( 
>
0) is the heat of adsorption and a is a proportionality constant 
(0
<
a
<
1) such that aq is the energy barrier which separates the adjacent adsorption 
sites. An important observation is that more strongly adsorbed molecules are less mobile 
than weakly adsorbed molecules [30] . 
In the case of surface diffusion, the gas concentration is well described by Henry ’ s law 
c = Kp , where K is the temperature dependent Henry ’ s law coeffi cient K = K
0
exp( q / RT ), 
where
K
0
is a proportionality constant
[29,30] 
. Since solubility is the ratio of the 
equilibrium concentration over pressure, the solubility is equivalent to the Henry ’ s law 
coeffi cient,
S
K
q
RT
s
=
⎛
⎝
⎞
⎠
0
exp
(5.10)
which implies that solubility is a decreasing function of temperature. 
The product of diffusivity and solubility gives
P
P
a q
RT
s
S
=
−
(
)
⎛
⎝
⎞
⎠
* exp
1
(5.11)
where
P
S
*
is a constant and since 0
<
a
<
1 the total permeability will decrease with 
increased temperature meaning that any increase in the diffusivity is counteracted by a 
decrease in surface concentration [30] .