CC BY
26
UDC 541.64:542.953.2
I. Ronova, A. Alentiev, M. Bruma, G. Zaikov
INFLUENCE OF CONFORMATIONAL RIGIDITY AND FREE VOLUME ON THE DIELECTRIC
AND MEMBRANE PROPERTIES OF THE POLYIMIDE FILMS
Keywords: polyimide films, swelling in supercritical CO2, free volume, porous morphology, gas transport parameters, dielectric
constant.
In this article the dielectric and membrane properties of different polyimide films have been investigated. We have shown that the transport parameters and permittivity of polyimides depends on the conformational rigidity and free volume. Also it was shown that the values of some polyimide's conformational parameters calculated under the assumption of free rotation in the absence of voluminous substituents are practically equal to the values found experimentally from hydrodynamic data.
Ключевые слова: полиимидные пленки, набухание в сверхкритическом СО2, свободный объем, пористая морфология,
параметры переноса газа, диэлектрическая постоянная.
В статье исследованы диэлектрические и мембранные свойства различных полиимидных пленок. Показано, что транспортные параметры и проницаемость полиимидов зависит от конформационной жесткости и свободного объема. Также показано, что значения конформационных параметров некоторых полиимидов, рассчитанных при условии свободного вращения при отсутствии объемных заместителей практически равны значениям, найденным экспериментально из гидродинамических данных.
1. Introduction
Polyimides represent one of the most important classes of high-performance polymers due to their high-temperature durability, good mechanical properties, and excellent chemical and thermal stabilities [1-4]. It is known that most of polyheteroarylenes, particularly polyimides, have high thermal and mechanical properties due to their conformational rigidity with Kuhn segment being in a wide range of values from 10 A to 15 A, which is characteristic for aliphatic polymers, to a thousand of Angstroms [5]. Polyimides are used in applications demanding service at enhanced temperatures while maintaining their structural integrity and a good combination of chemical, physical and mechanical properties.
Much of the research devoted to the development of high performance polymers for gas separation application has focused on the variation of chemical structure in order to obtain novel polymers with both high permeability and high selectivity [6]. Since a typical reverse relationship between permeability and selectivity exists, to make membrane separation more competitive, an important goal of polymeric membranes design is to develop materials which have both high permeability and selectivity. For such a purpose glassy polymers are used. Thus, various classes of polymers have been synthesized and studied for use as gas separation membranes. Among these, polyimides are in an important position since they exhibit extraordinary high gas selectivity as well as excellent thermal and mechanical stability, and film-forming ability [7, 8]. Besides, there is a broad possibility of varying the chemical structure of the repeating unit aiming to change the physical properties of polyimides, including their transport characteristics [9-12].
Some requirements to use these polymers for interlayer and intermetal dielectrics in advanced microelectronic applications are: high thermal stability, high glass transition temperature, good mechanical
properties, low dielectric constant, low coefficient of thermal expansion and low moisture absorption. The dielectric constant of polyimides depends mainly on the free volume of polymer matrix and on the polarizability and hydrophobicity of macromolecular chains.
Here, we present an investigation of thin films of some polyimides which were synthesized in m-cresol or in carboxylic acid medium such as benzoic acid or salicylic acid. The synthesis and general characterization of these polyimides were described previously [13, 14]. Some correlations are shown between the conformational rigidity parameters, such as free volume and characteristic ratio, and the glass transition temperature.
2. Calculation methods
2.1 Calculation of conformational parameters
The correlation between physical properties of polymers and conformational rigidity of their chains shows that the contribution of conformational rigidity to their properties is significant [15]. The conformational rigidity of a polymer can be estimated using different parameters, such as statistical Kuhn segment (Afr) and characteristic ratio (C.). Kuhn segment was calculated as under the assumption of free rotation by using the equation (1) [5].
lim
Ur2 >^ rnn
(1)
where <R > is mean square distance between the ends of the chain calculated for all possible conformations, n is the number of repeating units, lo is the contour length of a repeating unit, and L = nlo is the contour length of the chain, a parameter which does not depend on the chain conformation.
All the values of Kuhn segment were calculated with Monte Carlo method and the geometry of the repeating unit was assigned by using quantum-chemical method AM1 [16].
We also used another parameter of conformational rigidity named characteristic ratio C«,
which shows the number of repeating units in Kuhn segment, as shown by equation (2).
C = 4
l0
(2)
For some of the studied polymers, we also calculated the Kuhn segment values taking into consideration the hindrance of rotation, according to the method previously described [16, 17]. Most of these polymers did not have any hindrance of rotation, or their hindrance was too low, and it was neglected. Previously it was shown that the values of conformational parameters calculated under the assumption of free rotation in the absence of voluminous substituents are practically equal to the values found experimentally from hydrodynamic data [18].
2.2 Calculation of free, occupied, accessible and fractional accessible volume
In order to correlate the geometry of the repeating units of polymers with transport properties, the following parameters were calculated: van der Waals volume (Vw), free volume (Vf), occupied volume (Voce), accessible volume (Vacs), fractional accessible volume (FAV).
The occupied volume (Vocc) of a repeating unit is given by equation (3) as being the sum of the Van der Waals volume fVw) of the repeating unit and the volume of space around this unit that is not accessible for a given type of molecule of gas, which is named "dead volume" (Vdead). It is evident that the occupied volume of a repeating unit depends on the size of the gas molecule.
Vocc=(V„+Vdead) (3)
The accessible volume of a polymer (Vacs) is given by equation (4), where NA = 6.02^1023 is Avogadro's number, p is the polymer density, and Mo is the molecular weight of the repeating unit.
= 1 Na »[/occ
3CS ,,
P MD
(4)
However, more often is used the so-called fractional accessible volume (FAV), without any dimensions, which gives a better concordance with the coefficients of diffusion and of permeability, that is given by equation (5) [19, 20].
FAV = VacsT (5)
To calculate the Van der Waals and the occupied volume of the repeating unit, we used the quantum chemical method AM1 to refine the structure of the monomer unit [16]. The model of the repeating unit is a set of intersecting spheres whose coordinates of centers coincide with the coordinates of atoms and the radii are equal to the Van der Waals radii of the corresponding atoms.
Van der Waals volume (Vw) of the repeating unit is the volume of the body of these overlapping spheres. The values of Van der Waals radii were taken from the reference [21]. The model of the repeating unit was placed in a box with the parameters equal to the maximum size of repeating unit. By using the Monte
Carlo method we designated the number of random points m that fall into repeating unit and the total number of tests M. Their ratio is multiplied by the volume of the box, as seen in equation (6)
Vw = (m/M)Vb ox (6)
Then we calculated the dead volume. Since the molecules of O2, N2 and CO2 have ellipsoidal shape, we calculated the dead volume of the two spheres with radii corresponding to the major and to the minor axes of the ellipsoid. A number of 106 spheres with the radius of the gas was generated for each atom of the repeating unit. The result was a system consisting of a repeating unit, surrounded by overlapping spheres of gas. Then, the system was placed in the "box", similar to the one used in the determination of Vw, and random points were generated in the volume of the box [22, 23]. Thus, without making any assumptions about packing of the polymer chains in the glassy state, we could quickly calculate the Van der Waals volume, and the occupied and the accessible volumes.
The free volume (Vf) was calculated with the equation (7):
V = 1 _Na'K
P
Mn
(7)
The value Vf , thus calculated, shows the volume which is not occupied by the macromolecules in 1 cm3 of polymer film.
3. Experimental methods
3.1 Preparation of polymer films
The polyimides were synthesized by polycondensation reaction of an aromatic diamine with an aromatic dianhydride by traditional method using meta-cresol or benzoic acid and salicylic acid as solvent [13, 24], at high temperature to allow the complete imidization process and to exclude the cross-linking. The polycondensation reaction was run with equimolar quantities of diamine and dianhydride, at room temperature for 3 h, and then at 200°C for another 7 h. After cooling down to room temperature, the resulting viscous solution was poured in methanol to precipitate the polymer. The fibrous precipitate was washed with methanol and dried in vacuum oven at 100°C. These polymers showed good solubility in common solvents having low boiling point, such as chloroform and tetrahydrofuran, which are very convenient for film preparation.
The films, having the thickness usually in the range of 20-40 ^m, were prepared by using solutions of polymers in chloroform, having the concentration of 15 %, which were cast onto cellophane film and heated gently to evaporate the solvent. The films were carefully taken out of the substrate. To remove the residual m-cresol, the films were further extracted with methanol in Soxhlett apparatus, followed by heating in vacuum at 70°C for 3 days.
3.2 Measurement of glass transition temperature
The glass transition temperature (Tg) of the polymers was measured by differential scanning
calorimetry, with a DSC-822e (Mettler-Toledo) apparatus, by using samples of polymer films. The samples were heated at the rate of 10°C/min under nitrogen to above 300°C. Heat flow versus temperature scans from the second heating run was plotted and used for reporting the Tg. The middle point of the inflection curve resulting from the second heating run was assigned as the Tg of the respective polymers. The precision of this method is ±7-10°C.
3.3 Measurement of density
The density of polyimide films was measured by using the hydrostatic weighing method. The study was performed with equipment for density measurement and an electronic analytic balance Ohaus AP 250D from Ohaus Corp US, with a precision of 10-5g, which was connected to a computer. With this equipment we measured the change of sample weight (density) during the experiment, with a precision of 0.001 g/cm3 in the value of density. Ethanol and isopropanol were taken as liquids with known density. The studied polyheteroarylenes did not absorb and did not dissolve in these solvents, which for these polymers had low diffusion coefficients. The characteristic diffusion times were in the domain of 104 - 105 s, even for the most thin films studied here, which leads to higher times, of 1-2 order of magnitude, than that of the density measurement. This is why the sorption of solvent and the swelling of the film must have only insignificant influence on the value of the measured density. All measurements were performed at 23°C. The density was calculated with the equation (8):
Ps = Pi • Wa / (Wa - Wi)
(8)
where ps is density of the sample, Wa is the weight of the sample in air, Wl is the weight of the sample in liquid, p is the density of liquid. The error of the density measurements was 0.3 - 0.5 %.
3.4 Measurement of dielectric constant
For each polymer in this series, dielectric permittivity of polyimide films was measured by using Alpha High Resolution Dielectric Analyzer from Novocontrol-Germany, in the domain of frequencies from 10-3 to 106 Hz, and it was approximated at the frequency equal to zero to obtain the value of dielectric constant (eo).
3.5 Measurement of transport parameters
The transport parameters at 25 ± 3°C for He, O2, N2 and CO2 were measured using a mass spectrometric technique [25, 26] and barometric techniques on a Balzers QMG 420 quadrupole mass spectrometer (Liechtenstein) MKS Barotron [27], respectively. The upstream pressure was 0.8-0.95 at, and the downstream pressure was about 10-3 mm Hg for spectrometric method, while for barometric technique that pressure was in the range of 0.1-1 mm Hg; therefore, the reverse diffusion of penetrating gas was negligible. The permeability coefficients P were estimated using the formula: P = Js l/Dp, where Js (cm3 (STP)/cm2 9 s) is the flux of the penetrant gas
through 1 cm2 of the film; Dp (cm Hg) is the pressure drop on the film; l (cm) is the film thickness. The diffusion coefficient D was determined by using the Daynes-Barrer (time lag) method: D = l2/6h, where h (s) is the time lag. The solubility coefficient S was estimated as the ratio: S = P/D
4. Results and discussion
4.1 The dependence of dielectric constant of polyimides on conformational rigidity and free volume
Table 1 shows the chemical structure of the repeating units of the first series of the studied polyimides and some of their physical properties. Some of these polyimides contain methyl substituents on phenylene rings (2, 4 and 6) and other contain hexafluoroisopropylidene groups (5 and 6) in the main chain [28]. Figure 1 presents the dependence of glass transition temperature on Kuhn segment.
Table 1 - Structure and properties of the first series of polyimides
O
C
II O
C ii
O
Polymer X p (g/cm3) Tg (°C) lo (A) Afr (A) C" Vw (A3) Vf (cm3/g) So
1 - 1.200 200 32.32 25.43 0.787 625.359 0.2915 2.54
2 CH3 1.169 228 32.32 26.2 0.811 656.092 0.3056 2.11
3 - 1.398 - 24.53 28.03 1.143 426.696 0.1889 3.28
4 CH3 1.337 278 24.53 28.87 1.177 460.613 0.2085 2.84
5 - 1.546 275 22.37 28.65 1.281 488.651 0.1648 2.31
6 CH3 CH3 1.488 287 22.37 29.1 1.319 522.676 0.1793 1.99
CHj
"VJ^pW
ch,
o-k
-CII
O
CF,
-C-
CF,
-C-
3 - O , 4 - O , 5- CF, , 6 - CF,
p - density of polymer film; Tg - glass transition temperature; lo - contour length of a repeating unit; Afr - Kuhn segment; C„ - characteristic ratio; Vw - Van der Waals volume; Vf - free volume;so - dielectric constant
It can be seen that the behavior of polymers 1-6 is as usual: with increasing of Kuhn segment, from 25.43 to 29.1 A, the glass transition temperature increases, from 2000C to 2870C [5]. Since all the points of this dependence are situated on a straight line, described by equation y = 324 + 20.85x with a high correlation coefficient R = 99.17%, the glass transition temperature of polymer 3 can be calculated because it could not be measured experimentally. Thus, its calculated value is 262oC. The value of glass transition temperature of this polymer is well situated on the line showing the dependence of glass transition temperature on free volume (Fig. 2). This dependence in figure 2 shows that
n
1
2
with the increasing of free volume, from 0.1648 to 0.2915 cm3/g in polymers 5, 3, 1, and from 0.1793 to 0.3056 cm3/g in polymers 6, 4, 2, the glass transition temperature in general decreases because the higher free volume allows the molecular fragments to change their conformation and therefore an increased mobility in polymer matrix appears which leads to the decrease of glass transition temperature.
300
230-
260
O
" 240 ■
220
200
Y=-324+20.85x, R=99.17%
25
—[—
26
-r-
27
-r-
28
—[—
29
-1
30
A,, A
Fig. 1 - The dependence of glass transition temperature (Tg) on Kuhn segment (Afr), for the first series of polyimides
300 -I
280-
U
% 260-
240-
220-
200-
y=367.04-502.19, R=89.26%
0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 V[P cm3/g
Fig. 2 - The dependence of glass transition temperature (Tg) on free volume (Vf), for the first series of polyimides
Figure 3 presents the dependence of free volume on conformational rigidity. This dependence is linear, with a high correlation coefficient, 97.24%. It confirms that with increasing rigidity of polymer matrix, the free volume decreases.
When comparing Figs. 1, 2, and 3, a contradiction appears: with increasing rigidity, the glass transition temperature should increase and the free volume should decrease. However, by going from the polymers without any substituents (1, 3 and 5) to the polymers containing methyl substituents on phenylene rings (2, 4 and 6, respectively) not only the glass transition temperature increases, but also the free volume increases. This can be explained in the following way: the introduction of methyl substituents
0.32-|
0.30-
0.28-
O)
"i 0.26-
o
0.24-
0.22-
0.20-
0.18-
0.16-
y=0.498-0,253x, R=97,24%
-I-;-p-1-1-1-1-1-1-1-1-1-e-]
0.7 0,8 0.9 1.0 1.1 1.2 1.3 1.4
Fig. 3 - The dependence of free volume (Vf) on characteristic ratio C , for the first series of
X 7
polyimides
on phenylene rings in ortho-position towards the bond between phenyl and imide rings increases the hindrance of rotation and consequently the rigidity of the polymer increases [18]. The amount of conformational isomers and the mobility of one repeating unit towards the others decrease, and consequently the glass transition temperature increases. On another hand, the methyl substituents do determine a loosing of packing, but the polymer can not benefit of the increased free volume because the rotation hindrance does not allow the formation of all conformations which should occupy that free volume when heating the samples. This is why the glass transition temperature of polymers containing methyl substituents does not decrease, but it increases in each pair of these polymers: Tg values of polymers 2, 4, and 6, containing methyl substituents, are 228, 278, and 287oC, respectively, being higher than those (200, 262, and 275oC, respectively) of polymers 1, 3, and 5, which do not have such methyl substituents. Table 1 shows also the values of dielectric constant eo. For each polymer in this series, dielectric permittivity was measured at six frequencies, from 1 Hz to 100 kHz, and it was approximated at the frequency equal to zero to obtain the value of dielectric constant, as shown in figure 4. Figure 5 presents the dependence of dielectric constant of the studied polymers on characteristic ratio [23].
Fig. 4 - The dependence of dielectric permittivity (e) on frequency, for polymer 5 from the first series of polyimides
X X X X
хЛ/ 1/.X
Fig. 5 - The dependence of dielectric constant (so) on
characteristic ratio C„, for the first series of » 7
polyimides
It can be seen that the points are also situated in pairs. The values of so of the polymers containing methyl substituents are lower than those of the related polymers which do not contain methyl groups. By introduction of methyl substituents, the packing of the polymer becomes loose. However, the general dependence divides into two branches: polymers 1 and 2 are situated on the up-going part, while polymers 3-6 are situated on the down-going part of this diagram. The presence of hexafluoroisopropylidene bridge between imide rings increases the loosing effect of methyl groups in polymers 5 and 6.
It is known that fluorine atom has a highly electronegative effect [29]. Therefore, it is interesting to study the distribution of electron density in the repeating unit of macromolecules. We used the quantum-chemical method AMI to calculate the charges on atoms which constitute the repeating units of both polymers 5 and 6. The charge on fluorine in hexafluoroisopropylidene segment was -0.17 e, while the charge on each hydrogen atom of isopropylidene group was only -0.08 e. Similar repulsion can appear between hexafluoroisopropylidene groups and oxygen atoms of carbonyl groups in imide rings. It follows that in glassy state repulsion appears between fluorine atoms of hexafluoroisopropylidene groups which leads to the formation of supplemental microcavities in this polymer.
Now we examine the packing of fragments of two macromolecules: one containing hexafluoroisopropylidene and the other containing isopropylidene group. We suppose that oxygen atoms of ether groups are situated close to each other, at a distance equal to the sum of their Van der Waals radii, as shown in scheme below.
When packing, it is possible that hexafluoroisopropylidene groups of different chains are situated as shown in this scheme. Now, we make the modeling of their interaction to each other. We presume that the groups come out of the plane of this drawing, to meet each other and we compare the energy of repulsion which appears between hydrogen or fluorine atoms during this movement as a function of the distance between these atoms. The energy of repulsion E between the three atoms of one group and the three atoms of the other group is given by equation (9):
2.72A
2.72A
X
XX
\-X X
Qiii
(9)
where qi is the charge on hydrogen or fluorine atom, r is the distance between atoms. The sum is done at the same value of angle at which the two groups come out of the plane. Figure 6 presents the energy of repulsion between groups during their movement towards each other in the process when they come out of the plane.
At R^-l.sA-srid Ihe angle 54' E. „,.-03.963 <J,'nQl At R,=1.1TA and 1ha angla 88" E. ^-¿b.ivl kJjnra
v. desire
-1-
м
Fig. 6 - The dependence of repulsion energy on angle 9 between hexafluoroisopropylidene groups (line 1) and on the angle between isopropylidene groups (line 2). Van der Waals radius of fluorine atom is 1.5 A; Van der Waals radius of hydrogen atom is 1.17 A
In both cases, the groups can approach each other at a distance not shorter than the sum of Van der Waals radii of the corresponding atoms. When the distance between hydrogen or fluorine atoms is equal to the sum of Van der Waals radii of the corresponding atoms, the energy of repulsion in case of fluorine atoms (93.963 kJ/mol) is higher than in case of hydrogen atoms (25.312 kJ/mol). Similar repulsion can appear between hexafluoroisopropylidene groups and oxygen atoms of carbonyl groups in imide rings. It follows that in glassy state repulsion appears between fluorine atoms of hexafluoroisopropylidene groups which leads to the formation of supplemental microcavities in this polymer.
The increase of free volume of polymers containing methyl substituents leads to the decrease of dielectric constant compared with related polymers which do not contain such substituents as seen in Table 1.
4.2 The dependence of membrane properties of polyimides on conformational rigidity and free volume
Table 2 shows the chemical structure of the repeating unit, glass transition temperature Tg, density q, and conformational parameters Afr and Cw, calculated using Equations 1 and 2, of the second series containing 12 polyimides. These polyimides are based on common dianhydride and diamine monomers and their gas transport properties toward H2, CO, CO2 and CH4 have been published [30, 31]. We investigated the influence of conformational rigidity on the packing of macromolecules in glassy state and, consequently, on distribution of the free volume in polymer matrix. As seen in Table 2, the conformational rigidity, i.e., Kuhn segment, of the studied polyimides varies in a relatively large interval, from 22 to 44 A, and glass transition temperature from 190°C to 410°C.
Table 2 - Repeating units and properties of the second series of polyimides
Polymer Tg (oC) P. (g/cm3) lo (Ä) Afr (Ä) C«
Pi-1 230 1.292 30.97 35.89 1.16
Pi-2 250 1.390 30.97 36.97 1.19
Pi-3 192 1.252 42.05 24.10 0.573
Pi-4 240 1.336 32.63 22.90 0.702
Pi-5 205 1.260 32.24 28.04 0.870
Pi-6 230 1.331 52.51 34.25 0.652
copoly-
mer 1:1
Pi-7 210 1.262 75.25 28.98 0.385
copoly-
mer 1:1
Pi-8 230 1.290 32.05 27.51 0.847
Pi-9 282 1.330 22.41 24.2 1.08
Pi-10 305 1.349 17.22 25.80 1.498
Pi-11 410 1.400 17.67 43.9 2,48
PAI 197 1.259 17.40 36.84 2.12
Repeating unit Pi-1
CH3
O O
Pi-2
O O
O O
Pi-3
O CH3 O
Pi-4
-oar
CH3 0
Pi-5
O CH 0
OO
Pi-6 copolymer 1:1 -
O O
OO
0
Pi-7 Copolymer 1:1
CH3
I 3
O O
0 ch 0
Pi-8
Pi-9
0
o
Pi-10
0
C
-«no
O)
Pi-11
PAI
Thus, the selected series of polyimides is quite representative. The dependence of glass transition temperature (Tg) on rigidity (Afr) divides these selected polymers in three groups (Figure 7). Each of them behaves in a normal way: with increasing rigidity, the glass transition temperature increases [32]. The correlation coefficients in all these three groups of polyimides are high enough, in the range of 97.0699.99%.
O
n
O
O
O
O
O
O
O
CH
o
o
0
O
O
O
O
O
O
O
O
O
O
O
O
O
0
0
o
u
K
300 250 ■ 200 150-
y=-72.91+11.00x, R=99.99%
y=-264.01t22.178)i, R=98.24% 10
y =93.90+4.00x, R=97.06% 2
"3 5
20
—I—
25
—i-i-1—
30 35
A.,. A
—I—
40
—I—
45
Fig. 7 - The dependence of glass transition temperature (Tg) on Kuhn segment (Afr) for the second series of polyimides
The dependence of glass transition temperature (Tg) on free volume (Vf) is linear and situates all these polymers on one line with the exception of polymer PAI (Figure 8). The values of Vf are given in Table 3. The correlation coefficient is high enough, 96.53%. With increasing free volume of the polymer, the glass transition temperature decreases.
Higher the free volume, more conformational transitions can take place in polymer matrix at heating and the highly elastic state appears faster. At the same time, the free volume of the polymer is determined by its capacity to pack in glassy state and consequently it depends on its conformational rigidity.
450-,
400 -350300 -
u
Q
** 250 -200-
150
y=1118.105-3S32.464x, R=96.53%
"T "*' T '—r 1 r 1 X <—r '* 1—>~1 0.16 0,18 0,20 0.22 0.24 0.26 0.28 0.30 0.32 0.34
Vf, cm'lg
Fig. 8 - The dependence of glass transition temperature (Tg) on free volume (Vf) for the second series polyimides
Figure 9 shows the dependence of free volume (Vf) of the studied polyimides on conformational rigidity (Afr). It can be seen that there are three dependences (three lines), and in each of them enter the same polymers as in Fig. 7. The correlation coefficients of all three dependences are very high, in the range of 97.53-99.99%.
We examine now how the conformational rigidity influences on the permeability coefficients P of the membranes made from these polymers. For these polymers we calculated the Van der Waals volume, free volume, occupied volume, and accessible volume (Table 3). Table 4 shows the permeability coefficients
to four gases, H2, CO, CO2 and CH4, taken from publications [30, 31].
Fig. 9 - The dependence of free volume (Vf) on Kuhn segment (Afr) for the second series polyimides
Table 3 - Van der Waals (Vw), free (Vf), occupied (Voce) and accessible (Vacs) volumes of the second series of polyimides
№ Vw (A3 ) Vf (cm3/g) H2 CO
Vocc (A3 ) Vacs (cm3/g) Vocc (A3 ) Vacs (cm3/g)
1 610.900 0.2239 646.051 0.1919 660.039 0.1794
2 641.686 0.2220 679.993 0.1923 695.243 0.1804
3 825.166 0.2434 876.832 0.2088 899.047 0.1939
4 635.673 0.2263 672.665 0.1958 688.284 0.1830
5 625.359 0.2422 661.567 0.2103 676.362 0.1972
6 1024.818 0.2268 1087.410 0.1948 1108.728 0.1838
7 1436.066 0.2397 1522.883 0.2060 1559.086 0.1921
8 626.346 0.2339 660.248 0.2046 675.624 0.1987
9 426.696 0.2216 449.094 0.1937 458.160 0.1825
10 333.975 0.2153 349.409 0.1910 356.025 0.1806
11 333.975 0.1883 349.409 0.1640 356.025 0.1520
AI 325.572 0.2167 342.356 0.1870 349.811 0.1738
№ Vw (A3 ) Vf (cm3/g) CO2 CH4
Vocc (A3 ) Vacs (cm3/g) Vocc (A3 ) Vacs (cm3/g)
1 610.900 0.2239 652.406 0.1866 662.307 0.1777
2 641.686 0.2220 686.717 0.1817 697.961 0.1783
3 825.166 0.2434 886.774 0.2021 902.953 0.1913
4 635.673 0.2263 679.516 0.1902 691.053 0.1807
5 625.359 0.2422 668.369 0.2042 679.144 0.1947
6 1024.818 0.2268 1097.122 0.1898 1112.292 0.1820
7 1436.066 0.2397 1539.180 0.1997 1565.260 0.1897
8 626.346 0.2339 667.042 0.1987 678.383 0.1889
9 426.696 0.2216 453.330 0.1885 459.656 0.2402
10 333.975 0.2153 352.390 0.1863 357.052 0.1790
11 333.975 0.1883 352.390 0.1593 357.052 0.1520
AI 325.572 0.2167 345.713 0.1810 351.117 0.1714
Since the accessible volume Vacs is considered as the volume of the space inside the polymer which can be occupied by the centers of gas molecules so that the Van der Waals spheres of the gas should not overlap on the Van der Waals spheres of the atoms of polymer chain [33], one can approximate the free volume using the equation 10.
P = Aexp(-B/Vacc)
(10)
However, more often is used the so-called fractional accessible volume (FAV), without any dimensions, which gives a better concordance with the coefficients of diffusion and of permeability [19] that is
given by Eq. 5. Thus, the Eq. 10 can be written as shown in Eq. 11
P = Aexp(-B/FAV) (11)
Table 4 - The permeability coefficients (Barrer)
Figure 10 shows the dependence of permeability coefficients P of the studied polyimides on the fractional accessible volume FAV in the system one polymer-different gases. All these dependences are linear, with high correlation coefficients, being in the range of 99.05-99.91% (Table 5).
Table 5 presents the parameters (A and B) of Eq. 11 and the correlation coefficients (R). Parameter B means the slope of the line representing the dependence of permeability coefficients on the fractional accessible volume.
Table 5 - The slopes (A and B) and correlation coefficients (R) for curves described by equation (11) for the system one polymer - various gases
Polymer A -B R (%)
Pi-1 24.30 5.92 99.91
Pi-2 27.33 7.08 99.54
Pi-3 22.82 5.82 99.87
Pi-4 24.07 6.14 99.89
Pi-5 24.48 6.28 99.38
Pi-6 31.70 8.12 99.05
Pi-7 16.95 4.23 99.55
Pi-8 23.92 6.11 99.15
Pi-9 32.76 8.29 99.77
Pi-10 43.92 11.31 99.71
Pi-11 27.21 6.24 99.97
PAI 22.83 5.32 99.69
This parameter reflects the common selectivity of the respective polymer, which means that it shows how fast one gas penetrates through the polymer
membrane compared with another gas. Higher the value of B, better the selectivity of the respective polymer, which means that the respective polymer will better separate one gas from another one. This behavior is determined by the packing of the polymer in glassy state that is by the distribution of microcavities in polymer matrix. The packing of polymer is significantly determined by its conformational rigidity. In Table 5, it can be seen that polymers 6, 9, and 10 have the highest values of coefficient B, being 8.12, 8.29, and 11.31, respectively.
1/FAV
a
0.1-
O-
o
o
0.01 1
3.6 3.8 4.0 A. 2 4 .4 4.6 1JFAV
b
1/FAV
с
Fig. 10 - The dependence of permeability coefficients (P) on fractional accessible volume (FAV) in the system one polymer-different gases for the second series polyimides
Figure 11 presents the dependence of parameter B on characteristic ratio Cw. It can be seen that the selectivity of the polymer reaches the maximum when the characteristic ratio is 1.5, which means that Kuhn segment contains 1.5 repeating units. At such a value of Cw the packing of polymer in glassy state gives presumably a narrow distribution of microcavities and
№ H2 CO CO2 CH4
1 2.53 ±0.02 0.0567 ±0.0015 0.644 ±0.012 0.0329 ±0.0007
2 5.91 ±0.04 0.145 ±0.005 1.64 ±0.02 0.0504 ±0.0033
3 3.20 ±0.03 0.0678 ±0.0012 0.762 ±0.011 0.0305 ±0.0027
4 3.56 ±0.01 0.0836 ±0.0016 0.891 ±0.039 0.0418 ±0.0021
5 5.07 ±0.03 0.136 ±0.001 1.62 ±0.09 0.080 ±0.007
6 1.79 ±0.01 0.0395 ±0.0016 0.428 ±0.002 0.0099 ±0.0002
7 4.43 ±0.04 0.250 ±0.01 1.57 ±0.04 0.205 ±0.008
8 5.59 ±0.01 0.102 ±0.004 1.47 ±0.08 0.089 ±0.002
9 3.14 ±0.04 0.0367 ±0.0017 0.594 ±0.035 0.016 ±0.001
10 1.25 ±0.01 0.00397 ±0.00019 0.0627 ±0.0029 0.0011 ±0.0001
11 1.14 ±0.02 0.0176 ±0.0007 0.1971 ±0.0033 0.00795 ±0.00028
AI 1.63 ±0.01 0.0342 ±0.0005 0.435 ±0.004 0.0163 ±0.0003
the permeability of one gas is significantly higher than that of the other, which leads to high selectivity.
Fig. 11 - The dependence of slope B of equation (5) on characteristic ratio Cx for the second series of polyimides
On the other hand, as seen in the dependence of free volume on conformational rigidity, with increasing rigidity the free volume decreases, and in case of polymer PAI and Pi-11 it is the lowest, which means that the packing of macromolecules in these two polymers is moredense.
When we examine the dependence of permeability coefficients on fractional accessible volume for one gas and different polymers (Fig. 12), we can see that the points corresponding to polymers 6, 9, and 10 falls down visibly from the general dependences, for all four studied gases: CO2, CO, H2, and CH4.
0.01
y=4.589-1,163*. R-81,95
CO,
—I—■—I—■—I—'—I—■—I—■—I—'—I—'—I—
3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 1iFAV
0.1 -
0.01 г
1E-3-
10 .
y=3.383-1.085x, R=72.24%
CO
■—i—'—i—'—i—'—г—*—i—'—г—i—i—>—i— 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.б 4.7 1/fav
CL =
2 5
PAI
10"
y=4.678-1.060x, R=95.97%
.11
H„
—i—'—i—'—i—'—i—'—i—'—i—'—i—'—i—
3.7 3.8 3.9 4.0 4.1 4.2 4.3 4,4 1/FAV
С
0,1
Q-C
0.01
CH.
y=2.600-0.976x, R=S3.23%
i—|—i—|—i—|—i—|—i—|—i—|—i—|—.—|—. 4.0 4.1 4.2 4.3 4,4 4.5 4.6 4.7 4.8 1/FAV
d
Fig. 12 - The dependence of permeability coefficients (P) on fractional accessible volume (FAV), for the system one gas-various polymers for the second series of polyimides
It shows that even if the selectivity is high, the permeability of these polymers is low. The geometrical structure of the repeating unit determines the conformational rigidity of the polymer, its packing in glassy state, and subsequently its membrane characteristics: permeability and selectivity. Thus in this section we have shown that the transport parameters and permittivity of polyimides depends on the conformational rigidity and free volume.
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© I. Ronova - Leading Researcher, Doctor of Chemistry, Nesmeyanov Institute of Organoelement Compounds, Moscow Russia, A. Alentiev - Doctor of Chemistry, Professor, Topchiev Institute of Petrochemical Synthesis, Moscow, Russia, M. Bruma - Head of Laboratory, Doctor of Chemistry, "Petru Poni" Institute of Macromolecular Chemistry, Iasi, Romania, G. Zaikov - Doctor of Chemistry, Professor of Plastics Tecnology Department, Kazan National Research Technological Univercity, [email protected].
© И. Ронова - ведущий научный сотрудник, доктор химических наук, Институт элементоорганических соединений им. А.Н.Несмеянова РАН, Москва, Россия, А. Алентьев - доктор химических наук, профессор, Институт нефтехимического синтеза им. А.В.Топчиева РАН, Москва, Россия, М. Брума - заведующий лабораторией, доктор химии, Институт макромолекулярной химии им. Петру Пони, Яссы, Румыния, Г. Заиков - доктор химических наук, профессор кафедры Технологии пластических масс, Казанский национальный исследовательский технологический университет, Казань, Россия, [email protected].
