Membrane Gas Separation

7.2.2 Modelling Gas Solubility Into Mixed Matrix Glassy Membranes 
Frequently, gas solubility in a composite matrix is represented through a simple additive 
rule that considers the sorption capacity of the polymer and the fi ller unaffected by the 
mixed matrix conditions. Unfortunately this is not the case for mixed matrices obtained 
by loading impermeable nano - particles in high free volume glassy polymers, where the 
addition of the inorganic rigid phase affects essentially the actual density of the polymer 
and its swelling behaviour. The FFV increase is reasonably associated to the generation 
of defects at the polymer/fi ller interface, which on the average results in a decreased 

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Vapor Sorption and Diffusion in Mixed Matrices Based on Tefl on® AF 2400 
129
density of the polymer phase of the composite. Thus the presence of fi ller essentially 
affects only the polymeric phase, by adding free volume elements at the organic/inorganic 
interface, while the adsorption capacity on the fi ller surface can be considered practically 
unchanged, in view also of the fact that the adsorption on the fi ller surface is rather small 
compared to the total solubility. 
In general we can write the total solubility of a penetrant as a sum of two contributions, 
one due to the polymeric phase and one due to the fi ller; per unit volume of the mixed 
matrix one has:
C
n
n
V
C
C
i M
i F
i P
F
i F
F
i P
,
,
,
,
,
=
+
=
⋅
+ −
(
)⋅
Φ
Φ
1
(7.9)
where  
Φ
 
F
is the volume fraction of the fi ller in the composite, C
i,M
is the gas solubility 
per unit volume of mixed matrix, C
i,F
and C
i,P
are the gas solubilities in the fi ller and in 
the polymer, per unit volume of fi ller and per unit volume of polymer, respectively. 
(Indeed, on the fi ller particles only surface adsorption takes place so that the fi ller con-
tribution should be more properly written using the surface concentration  
γ
 
i
and the area 
per unit volume of the particles, A 
′
′
′
= 6/ d
p
for spheres of uniform diameter d
p
, so that 
C
i,F
=  
γ
 
i
A 
′
′
′
. However, since the particles here used are reasonably monodisperse we can 
equivalently use C
i,F
in place of  
γ
 
I
A 
′
′
′
.) Since the contribution due to adsorption is minor 
if not negligible, it appears reasonable to accept that the adsorption capacity of the fi ller 
in the composite is practically the same as onto the pure free particles,
C
i F
,
0
; thus we now 
assume that:
C
C
i F
i F
,
,
=
0
so that Equation (7.9) becomes:
C
C
C
i M
F
i F
F
i P
,
,
,
=
⋅
+ −
(
)⋅
Φ
Φ
0
1
(7.10)
Normally the major contribution to the solubility in the mixed matrix under consideration 
is given by the glassy polymer phase and thus by the second term in the right - hand side 
of Equation (7.10) . Therefore, if one can measure reliably the density
ρ
2
0
of the glassy 
polymer phase of the mixed matrix one can rely on the NELF model to calculate the 
solubility C 
i,P
  in the low pressure range or in the entire isotherm in the case of non -
swelling penetrants. For swelling penetrants also the swelling coeffi cient k
sw
, or the 
dilation isotherm of the polymer phase, would be required. Unfortunately neither
ρ
2
0
nor 
k
sw
values are commonly available for the polymer phase of a mixed matrix, and the 
limited existing data indicate that the polymer density in mixed matrices require 
techniques more delicate than usual to be determined, in the absence of which its value 
is obtained with an uncertainty too high to allow a direct use in the calculations of solu-
bility as indicated above. 
In the absence of a reliable value of the volumetric properties of the glassy polymer 
phase of the mixed matrix we can tackle the problem following a different procedure, 
according to which the required volumetric properties are obtained from the solubility 
isotherm of a penetrant, arbitrarily chosen as reference. We can thus select a convenient 

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130
Membrane Gas Separation
test penetrant and measure directly its adsorption onto the pure fi ller and its solubility 
isotherms in the mixed matrix as well as in the pure unloaded polymer. Then we can 
calculate the solubility isotherm in the polymer phase alone of the MMM, by means of 
Equation (7.10) . Use of the NELF model allows one to calculate the unpenetrated polymer 
density  
ρ
 
2
at any pressure, considering for the binary parameter  
Ψ
  the same value valid 
for the unloaded polymer phase. From the dilation isotherm thus calculated for the polymer 
phase alone we obtain
ρ
2
0
as well as the swelling coeffi cient k
sw
in the mixed matrix. The 
value of
ρ
2
0
represents the pure unpenetrated polymer density in the mixed matrix, which 
obviously holds true also when other penetrants are used in the same MMM. The value 
of k
sw
accounts for the swelling effects of the specifi c penetrant on the polymer and thus 
it may vary for every polymer – gas couple. The value of
ρ
2
0
thus obtained may then be 
used to calculate through the NELF model the solubility of all other penetrants, at least 
in the low pressure range in which swelling is negligible. In the presence of important 
swelling effects one needs to determine also the swelling coeffi cient k
sw
of the polymer 
phase in the MMM, which requires the knowledge of one solubility value of that penetrant 
at relatively high pressures, according to the procedure illustrated in a previous work [7] . 
The determination of
ρ
2
0
as described above allows us also to evaluate the fractional free 
volume in the unpenetrated polymer phase, commonly defi ned as:
FFV
V
V
V
W
W
W
=
−
⋅
=
−
⋅
2
2
2
2
0
2
1 3
1 3
.
.
ρ
ρ
ρ
(7.11)
where V
W 
is the van der Waals volume of the repeating unit of the polymer, whose value 
can be calculated based on Bondi ’ s group contribution method [15] , and is already avail-
able for the matrices of interest in this study [4] .