Membrane Gas Separation

d
B
=
(
)
λ
θ
2 sin
(3.6)
The fi rst value of
d
B
in PTMSN can be attributed to intersegmental spacing and is 
indicative of looser chain packing as compared with other presented norbornene 
polymers. 
It is worthwhile to note that the conventional glassy polymers like polycarbonates and 
polysulfones are characterized by much smaller values of d spacing (4 – 6 Å ); that is, they 
have more densely packed chains. 
The generally recognized and the most reliable method for investigation of free volume 
in polymers is positron annihilation lifetime spectroscopy (PALS). It was applied for 
investigation of PTMSN and related polymers. This method is based on the measurement 
of lifetime spectra of positrons in polymers – lifetimes  
τ

i
  (ns) and corresponding intensi-
ties I 
i
  (%). Longer lifetimes  
τ

i
  (or  
τ
 
3
and  
τ
 
4
) (so - called o - orthopositronium lifetimes) can 
be related to the mean size of free volume R .
τ
π
π
i
i
i
R R
R R
=
−
(
)
+
( )
(
)
[
]
−
1 2 1
1 2
2
0
0
1
sin
(3.7)
where  
τ

i
  =  
τ
 
3
or  
τ
 
4
are o - Ps lifetimes and R 
i
  = R
3
or R
4
are the radii of free volume ele-
ments expressed in nanoseconds and angstroms, respectively;
R
0
= R 
i
  +  
Δ
 R , where 
 
Δ
 R = 1.66 Å is the fi tted empirical parameter. Earlier studies
[8,9] 
with other poly-
norbornenes with H and CH 
3
substituents revealed relatively short lifetimes  
τ
 
3
and smaller 
sizes of free volume elements in these polymers. Table
3.9 
presents the PALS data 
Table 3.8  Fractional free volume ( FFV ) and X - ray scattering data of addition - type 
polynorbornenes with different side groups R
R
FFV
WAXD
Ref.
(2  
θ
  ) 
1
, degree
(2  
θ
  ) 
2
, degree
d 
B1
, Å
d 
B2
, Å
Si(CH 
3
) 
3
0.275
6.5
15.5
13.6
5.7
This work
CH 
3
0.18
9.3
18.2
9.3
4.9
[9]
H
0.15
10
18.5
8.8
4.7
This work, [8]
Table 3.9 PALS data for PTMSN and other addition - type polynorbornenes 
R
 
τ
 
3
, ns
I 
3
, %
 
τ
 
4
, ns
I 
4
, %
R 
3
/R 
4
, Å
Ref.
Si(CH 
3
) 
3
3
10
7
30
3.7/5.7
This work
CH 
3
3.15
33.3
–
–
3.34 (R 
3
)
[9]
H
2
25
3.6
8.9
2.9/4.1
This work

---

Addition-type Polynorbornene with Si(CH
3
)
3
 Side Groups
53
obtained for the PTMSN investigated. First, the treatment of the primary experimental 
results shows that much better statistical fi t can be obtained for bimodal lifetime and size 
distribution of free volume for PTMSN. According to Equation ( 3.7 ), very large free 
volume elements are present in this polymer. The values 2 R
4
are consistent with the results 
of WAXD study (Table 3.8 ). Second, it should be reminded that bimodal lifetime distri-
bution observed for PTMSN is a common feature of all polymers with great gas perme-
ability [19] .
An analysis of kinetics of o - positronium formation and decay in the polymer allows 
an estimation of the concentrations of free volume elements of the two kinds in PTMSN: 
N
3
= 5.6
×
10 
19
cm 
– 3
and N
4
= 9.8
×
10 
19
cm 
– 3
. For comparison in PTMSP the correspond-
ing values are somewhat higher: N
3
= 5.6
×
10 
19
cm 
– 3
, N
4
= 26
×
10 
19
cm 
– 3
. A big difference 
in the N
4
values can explain more opened porosity and other properties of PTMSP. 
Temperature dependence of the PALS parameters in PTMSN is shown in Figure 3.5
for a wide range of temperature. Strong independence of the observed PALS parameters 
on temperature is rather unusual for common polymers, however, it has been observed 
for high free volume materials like PTMSP [29] . Usually temperature dependence of 
lifetimes and radii of free volume can be described by two linear dependencies with dif-
ferent slopes and a break in vicinity of the glass transition temperature [30] . In conven-
tional glassy polymers, increases in the lifetimes or hole size with temperature are caused 
mainly by small scale movements of groups that form ‘ walls ’ of free volume elements in 
polymers. In extra high permeability polymers like PTMSN and PTMSP, the sizes of 
holes (and corresponding lifetimes) are determined mainly by loose packing of rigid 
chains, so this small scale mobility is masked by structure porosity frozen in polymer.
Inverse gas chromatography provides an additional possibility to determine the size of 
free volume elements in glassy polymers. For several highly permeable glassy polymers, 
it was shown that the dependencies of the partial molar enthalpies of mixing  
Δ
 H
m
are 
strong functions of molecular sizes of solutes and pass through minima that correlate with 
the gas permeability [19] . So it is worthwhile to use the size of solutes, e.g. the critical 
volume V
c
, as a scaling parameter and compare the coordinates of these minima for dif-
ferent glassy polymers. It can be assumed that the molecular size of the solute, which 
Measurement temperature, ºC
5
100
150
2
3
4
5
6
7
8
10
20
30
40
Intensity (%)
τ
4
I
4
τ
3
I
3
Lifetime (ns) 
Figure 3.5 Temperature dependence of the PALS parameters in PTMSN

---

54
Membrane Gas Separation
corresponds to the minimum at this dependence, is close to the mean size of free volume 
elements in the polymer. So it was interesting to check this for PTMSN and make some 
comparisons. Figure 3.6 shows the correlation of the  
Δ
 H
m
values of PTMSN with the size 
of the solutes ( n - alkanes) whose size is characterized by V
c
.
One can discern for PTMSN the two broad minima in the curve  
Δ
 H
m
( V
c
) that corre-
spond to the values of V
c
of 426 cm 
3
/mol (n - C 
7
) and 754 cm 
3
/mol (n - C 
12
). The presence 
of the two minima is rather unexpected because in most cases only one minimum was 
observed in the curves  
Δ
 H
m
( V
c
). The only exception so far was amorphous Tefl on AF2400, 
where also two minima, though fl at and broad, could be noted [19] . Here we have appar-
ently another manifestation of bimodal size distribution of free volume in glassy polymers 
that attracts now a keen interest [33] . The error bars shown in Figure 3.6 seem to support 
the assumption of bimodal size distribution in this polymer. 
The task of the quantitative determination of the free volume size using the IGC method 
must depend on a choice of molecular volume of n - alkanes, the solutes used by us as the 
probes in free volume determination. Different scales can be employed as measures of 
molecular volumes: van der Waals volume V
w
, molecular volume in liquid phase V
b
(at 
corresponding boiling point T
b
) and V
c
. These quantities vary signifi cantly and differ from 
each other; however, the general trends are the same for series of the solutes: for example 
for n - hexane the following values are reported(in Å
3
/molecule): V
w
= 113, V
b
= 235 and 
V
c
= 611 (the sources of these values is given in Ref. [34] ). Since the density of the sorbed 
phase is unknown, the only criterion for selection the of best scaling parameter should be 
a comparison with the results of other probe methods for the investigation of free volume 
in polymers. 
In Table 3.10 , the sizes of free volume elements ( V
f
, Å
3
) in PTMSN and other polymers 
are compared. They were found via PALS and IGC. The pairs of V
f
values for PTMSN 
based on the IGC method correspond to the two minima shown in Figure 3.6 .
Δ
H
m
, kJ/mol
10
0
–10
–20
–30
100
300
500
700
900
1100
V
c 
cm
3
/mol
C
12
C
3
C
10
1
2
3
Figure 3.6 Correlation of  
Δ
H
m
versus V
c
 : (1) PTMSN; (2) PVTMS [31] ; (3) AF1600 [32]

---

Addition-type Polynorbornene with Si(CH
3
)
3
 Side Groups
55
The PALS values correspond to the larger size of microcavities in the case of the 
bimodal size distribution. In addition, the permeability of the polymers in respect of 
oxygen is also shown. It is seen that the polymers with greater gas permeability are char-
acterized by larger free volume elements independently of the method of the determina-
tion. However, the values obtained by the IGC method differ systematically from those 
found using the maybe more refi ned PALS method. In particular, V
f
(PALS) are smaller 
than V
f
( V
c
) but larger than V
f
( V
b
) with approximately equal deviation of V
f
(PALS) from 
the both quantities. There are several different reasons for these discrepancies. In 
the PALS method, the radii of free volume elements are estimated via o - positronium 
lifetimes. The size of free volume elements V
f
is calculated then in assumption of its 
spherical symmetry, which is the subject of some doubts. In the case of IGC, the sizes of 
the free volume element are averaged over the temperature range, where  
Δ
 H
m
values were 
determined, while the PALS method gives the information at specifi c temperatures. In 
addition, as has been mentioned, the main uncertainty in estimation of V
f
in the IGC 
method is related to the density of sorbed phase and, consequently, the best choice for 
the scaling parameter – V
b
or V
c
. It can be speculated that the critical density, which 
defi nes V
c
, is too small for the condensed phase. On the other hand, the conformations of 
the sorbed probe molecules (n 
- 
alkanes) can poorly correspond to irregular structure 
of free volume in polymers, so their accommodation can be rather imperfect. Since the 
deviations of V
f
( V
c
) and V
f
( V
b
) from V
f
(PALS) are more or less the same for all the 
polymers considered, one can assume that the density of the sorbed phase is larger than 
the critical density, which defi nes the V
c
values, by a factor about 2. This conclusion can 
be taken into account in further determination of the sizes of free volume elements using 
the IGC method.