Membrane Gas Separation

7.2.1
NELF Model 
The NELF model [6 – 9] enables us to calculate the solubility isotherms in glassy polymers, 
on the bases of the results of non - equilibrium thermodynamics of glassy polymers [9] . It 
offers an explicit relationship between penetrant fugacity on one hand and penetrant solu-
bility in the glass and glassy polymer density on the other hand. Thus it accounts for the 
strong and important relationship existing between solubility and fractional free volume 
in the glassy polymeric phase. In particular, the NELF model is an extension of the lattice 
fl uid equation of state [10] to the non - equilibrium state that characterizes glassy polymers. 
The model uses the same characteristic parameters ( p * ,  
ρ
  * , T * ) of the Sanchez and 
Lacombe theory [11] to evaluate the properties of pure components, and the same mixing 
rules to estimate the mixture properties. 
The characteristic parameters of the polymer can be calculated by best - fi tting the LF 
equation of state to PVT data above T
G
, while for the penetrant either PVT or vapour –
liquid equilibrium data can be profi tably used. 
The number of lattice sites occupied by a molecule in its pure phase is given by:
r
P M
RT
M
v
i
i
i
i
i
i
i
i
0
=
=
*
* *
* *
ρ
ρ
(7.3)
where M 
i
  is the molar mass of component i and
v
i
*
is the volume occupied by a mole of 
lattice sites of pure substance. The parameter
r
i
0
is usually set to infi nity for the polymer 
species. Mixing rules with only one adjustable binary parameter,
 
Ψ
  
, are adopted to 
calculate the mixture properties.  
Ψ
  affects the binary characteristic pressure
P
12
*
that rep-
resents the energetic interactions per unit volume between gas and polymer molecules:
P
P
P
12
11
22
*
*
*
=
⋅
Ψ
(7.4)
A fi rst - order approximation is given by  
Ψ
  = 1, to use when no specifi c data for the mixture 
are experimentally available. 
The pseudo - equilibrium state of the glassy mixture is described by the usual state vari-
ables (temperature, pressure and composition) plus the polymer density  
ρ
 
2
that accounts 
for the departure from equilibrium frozen into the glass. 
The phase equilibrium between the external gas phase and the glassy polymer mixture 
is given by the usual relationship:

---

128
Membrane Gas Separation
μ
ρ
μ
1
1
2
1
NE s
PE
PE
Eq g
( )
( )
(
)
=
(
)
T p
T p
, ,
,
,
Ω
(7.5)
where the penetrant chemical potential in the solid non 
- 
equilibrium phase,
μ
1
NE s
( )
, is 
defi ned as:
μ
ρ
1
1
2
2
NE s
tot
( )
= ∂
∂
G
n
T p n
, ,
,
(7.6)
and is given by an explicit expression [6 – 8] according to the Sanchez - Lacombe equation 
of state. Superscripts PE in Equation (7.5) label pseudo - equilibrium conditions. 
In order to evaluate solubility isotherms through Equation (7.5) one thus requires the 
knowledge of the characteristic parameters of the pure components that may be found in 
specifi c collections, as well as of the actual density of the glassy polymer at the actual 
experimental conditions, that depends also on the thermo 
- 
mechanical history of the 
samples and can be determined from separate direct dilation experiments. For non 
-
swelling penetrants, like many permanent gases such as O 
2
, N 
2
and CH 
4
, one can use the 
pure unpenetrated polymer density
ρ
2
0
for the entire isotherm. For swelling penetrants the 
value
ρ
2
0
for the polymer density can be used only in the low pressure range [7] , while 
for the entire isotherm one needs to account for the actual polymer density given by the 
dilation isotherm which parallels the penetrant sorption isotherm. It has been shown that 
in the cases experimentally tested [12 – 14] there is a linear dependence of the polymer 
density on the partial pressure of the vapour sorbed and thus we can introduce a swelling 
coeffi cient, k 
sw 
, to account for such a linear relationship between polymer density and 
penetrant pressure during sorption:
ρ
ρ
2
2
0
1
PE
sw
p
k p
( ) =
−
(
)
(7.7)
Since specifi c dilation isotherms are not usually available in open literature data, use of 
Equation (7.7) allows reasonable and simple procedures: in the absence of dilation iso-
therm data, one single solubility datum at high pressure can be used in Equations (7.5)
and (7.7) to obtain the swelling coeffi cient k
sw
, so that the swelling isotherm of the matrix 
at all other pressures can then be obtained through Equation (7.7) . 
Using Equations (7.5) and (7.7) , the phase equilibrium condition becomes of the fol-
lowing type:
μ
ρ
μ
1
1
2
0
1
NE s
PE
sw
Eq g
T p
k
T p
( )
( )
(
)
=
(
)
, ,
,
,
,
Ω
(7.8)