Membrane Gas Separation

 [volume fraction of PEG]
Density [g/cm
3
]
0,115
0,12
0,125
0,13
0,135
0,14
0,145
0,15
FFV
Experimental density
Theoretical density
FFV
Figure 12.4 Density and fractional free volume of Pebax
 
®
 
/PEG as function of volume 
fraction of PEG determined by additive model and buoyancy method [modifi ed plot from 
ref. 24 ]

---

236
Membrane Gas Separation
Figure 12.5 o - Ps lifetime and o - Ps intensity as a function of PEG content [reprinted with 
permission from ref. 75 ]
These results encouraged us to study in more detail this kind of blends, and the dem-
onstration of free volume increase by using PALS was our goal. If we could demonstrate 
it for rubbery - like membranes, novel strategies for improving the membrane performance 
could be developed. The success of application of PALS to polymers is largely due to 
the so - called standard model developed by Tao and Eldrup which allows a quantitative 
description of the relation between free volume and ortho - positronium (o - Ps) lifetime 
(  
τ
 
o - Ps
) [77,78] . This quantum mechanical model is based on an assumption that o - Ps is 
confi ned in spherical holes (free volume elements) with infi nitely high walls and gives a 
direct relationship between  
τ
 
o - Ps
and the size of the free volume:
τ
π
π
o-Ps
=
−
+
⎛
⎝⎜
⎞
⎠⎟ +
+
⎛
⎝⎜
⎞
⎠⎟
⎡
⎣⎢
⎤
⎦⎥
−
1
2
1
1
2
2
0
0
1
R
R
R
R
R
R
sin
(12.7)
where
R 
is the average radius of holes (free volume element),
R
0
= R +  
Δ
 R and 
 
Δ
 R = 0.166 nm. 
PALS gives information about size and size distribution of free volume in amorphous 
polymers. In interpreting the results on free volume variation in rubbery polymers we use 
the concept of dynamic free volume [75,79] . The results show that in these membranes, 
the FFV varies by addition of PEG (Figure 12.4 ), but the average size of the free volume 
element does not change signifi cantly at room temperature (Figure 12.5 ). As expected, 
this is due to the rubbery or apparent molten state of PEO. Although the size of these 
holes did not signifi cantly increase, the dynamic free volume has been greatly enhanced 
because of plasticizer behaviour of PEG. Thus the total free volume was increased and 
consequently the observed CO 
2
permeability was higher.

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Tailoring Polymeric Membrane Based on Segmented Block Copolymers
237
7.5
7.6
7.7
7.8
7.9
8.0
10
-7
10
-6
10
-5
CO
2
H
2
N
2
CH
4
n-C
4
H
10