Membrane Gas Separation

3.3.4
Free Volume 
A description of high permeability polymers is not complete if the free volume of the 
material is not considered. In this work, several methods for evaluation of free volume 

---

Addition-type Polynorbornene with Si(CH
3
)
3
 Side Groups
51
are used. All of them show that the appearance of bulky Si(CH 
3
) 
3
side groups results in 
a signifi cant increase in free volume. The simplest and, maybe, the most traditional 
approach for estimation of free volume (FFV) is based on the Bondi method [28] :
FFV
f
sp
w
sp
=
= −
V V
V V
1 1 3
.
(3.5)
where V
f
(cm 
3
/g) is the free volume, V
sp
(cm 
3
/g) is the specifi c volume of the polymer 
defi ned as 1/  
ρ
  where  
ρ
  (g/cm 
3
) is the density, and V
w
is the van der Waals volume of the 
repeat unit. This is a very approximate approach, however, as it gives systematic scales 
of glassy polymers with various density of chain packing, and the FFV values usually 
correlate with the permeability and diffusion coeffi cients. It is seen from Table 3.8 that 
PTMSN has higher values of FFV than other addition - type polynorbornenes. The found 
S
35
, cm
3
/cm
3
atm
10
8
10
7
10
6
10
5
10
4
10
3
10
2
10
1
10
–1
10
0
0
10
20
30
T
c
2
*10
–4
, K
C
2
C
15
2
1
40
50
60
Figure 3.4 Correlation of S (35 ° C) versus T
c

2
 , where T
c
is the critical temperature of 
solutes: (1) PTMSN; (2) PTMSP [22]
Table 3.7  Permeability, diffusion and solubility coeffi cients of gases in PTMSN and 
 PTMSP [27]
a

Gas
PTMSN
b
PTMSP
c
P
D /10 
6
S
P
D /10 
6
S
O 
2
838
5.0
1.3
7730
35.8
1.6
N 
2
321
2.7
0.91
4970
25.5
1.5
CO 
2
4180
3.2
9.8
28 000
21.7
9.8
CH 
4
849
1.8
3.6
13 000
22.7
4.4
a
P in Barrer, D in cm 
2
s 
−
1
, S in cm 
3
(STP)/cm 
3
atm.
b
At 22 ° C.
c
At 30 ° C.

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52
Membrane Gas Separation
absolute FFV value is comparable to those earlier observed for other highly permeable 
polymers like e.g. perfl uorinated copolymers AF [21] .
X - ray wide angle scattering in glassy polymers usually shows one or two broad amor-
phous halos. Most high free volume polymers are characterized by two bands, while e.g. 
polyimides show a single halo. The coordinates of the maximum of 2  
θ
  allow an estima-
tion of periodicity d
B
(so - called d spacing) using Bragg ’ s formula: