Free Volume and Transport Properties of Barrier and Membrane

Chapter 21

Free Volume and Transport Properties of Barrier and Membrane Polymers 1

2

B. D. Freeman and Anita J. Hill 1

Department of Chemical Engineering, North Carolina State University, Raleigh, NC 27695-7905 CSIRO Manufacturing Science and Technology, Private Bag 33, South Clayton MDC, Clayton, Victoria 3169, Australia

Downloaded by UNIV OF QUEENSLAND on October 6, 2015 | http://pubs.acs.org Publication Date: January 28, 1999 | doi: 10.1021/bk-1998-0710.ch021

2

Ultrahigh permeability polymers are stiff chain glassy polymers that pack very poorly in the solid state, having high values of fractional free volume and a high degree of connectivity of free volume elements. High barrier materials are prepared from stiff chain glassy polymers that pack efficiently in the solid state, with low fractional free volume values. Manipulation of solid state chain packing changes permeability coefficients over many orders of magnitude. Control of chain chemistry and packing structure of stiff chain glassy polymers permits rational tailoring of permeation properties between those of high barriers and those of extremely permeable membranes via systematic manipulation of free volume and free volume distribution. The transport of small molecules in polymers plays a key role in the use of membranes for liquid, gas, and vapor separations; barrier plastics for packaging; controlled drug delivery devices; monomer and solvent removal from formed polymers; and in the study of physical aging of glassy polymeric materials. In applications ranging from gas separation to barrier packaging, glassy polymers have permeation and separation characteristics superior to those of rubbery polymers and, as a result, glassy polymers are used commercially in these applications. Very high permeability membranes and low permeability barrier films derive from stiff chain, glassy polymers. High permeability membranes result from rigid disordered materials, and low permeability barrier films are fabricated from rigid locally ordered materials. For example, poly(1-(trimethylsilyl)-1-propyne) [PTMSP], a stiff chain polymer with a glass transition temperature in excess of 300°C, is the most permeable polymer known (1). Its oxygen permeability coefficient at ambient conditions is approximately 1 x l 0 cm (STP) cm/(cm s cm Hg), which is more than ten times the oxygen permeability of highly flexible, rubbery poly(dimethylsiloxane) [PDMS], the most permeable rubbery polymer (2-5). Bulky substituents and double bonds along the PTMSP backbone lead to rigid, twisted polymer chains which pack very poorly, resulting in extraordinarily high solubilities, diffusivities, and, in turn, high permeabilities. In complementary contrast, glassy liquid crystalline polymers [LCPs] such as poly(p-phenyleneterephthalamide) [PPTA] and poly(phydroxybenzoic acid-co-6-hydroxy-2-naphthoic acid) [HBA/HNA] pack -6

306

3

2

©1998 American Chemical Society

In Structure and Properties of Glassy Polymers; Tant, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1999.

307

very efficiently in the solid state and exhibit extremely high barrier properties. For example, at 35°C the oxygen permeability of HBA/HNA73/27, a random copolymer containing 73 mole percent HBA, and PPTA are 47xl0" and 80xl0" crn^STP) cm/(cm s cm Hg), respectively (6-77). These values are eight orders of magnitude lower than the oxygen permeability coefficient of PTMSP, demonstrating the extraordinary range of permeation properties between those of low free volume glassy polymers, which enjoy efficient chain packing in the solid state, and high free volume glassy polymers, where chain packing in the solid state is strongly frustrated. Free volume is a convenient concept to characterize the amount of space in a polymer matrix that is not occupied by the constituent atoms of the polymer and is available to assist in the molecular transport of small penetrant molecules. As noted by Adam and Gibbs (72), the concept of free volume embodies inter- and intramolecular interaction as well as the topology of molecular packing in the amorphous phase. In this regard, transport properties of polymers have been described by Peterlin (75) as having static and dynamic free volume contributions, thereby acknowledging the importance of packing-related free volume as well as cooperative segmental chain dynamics to penetrant transport. In developing new polymers for membrane and barrier applications and in the systematic processinginduced manipulation of permeation properties of polymers, free volume is often used to rationalize experimentally observed structure/property relations. Therefore, the relationship among polymer backbone structure as well as higher order structure, such as nematic order in LCPs, and free volume and free volume distribution in the solid state is important. In this chapter, we examine the effect of chain packing on permeation properties for three families of polymers that span the range from extremely low permeability, liquid crystalline barrier materials to intermediate permeability, amorphous gas separation membrane materials to ultrahigh permeability polymers. All of the polymers discussed are glasses at the ambient measurement conditions. Chain packing in these systems is characterized by density-based estimates of free volume and by positron annihilation lifetime spectroscopy [PALS], which permits an estimate of both the size and concentration of free volume elements in a polymer matrix. The effect of free volume on permeability of a range of penetrants of different sizes is presented. For the highest permeability glassy polymers, subtle variations in free volume distribution result in extraordinary changes in permeation properties. 15

15


2

Background The permeability of a polymer film of thickness i to a penetrant, P, is (74): P =- ^ - ,

P2-P1

(1)

where Ν is the steady state gas flux through the film, and p2 and pj are the upstream and downstream penetrant partial pressures, respectively. When the downstream pressure is much less than the upstream pressure, permeability is often written as (14): P = SxD,

(2)

where S, the apparent solubility coefficient, is the ratio of the dissolved penetrant concentration in the upstream face of the polymer to the upstream penetrant partial pressure in the contiguous gas or vapor phase, and D is the concentration averaged penetrant diffusion coefficient (14).


308

Permeability properties are very sensitive to chain packing in the solid state. Chain packing is often characterized in terms of free volume. However, a simple, direct, unambiguous measurement of free volume in polymers is not available. The most common characterization of solid state chain packing is fractional free volume, FFV, which is defined as follows (14): FFV =

(3)

V

In this expression, V is the polymer specific volume, and V is the so-called occupied volume of the polymer, which is commonly estimated as 1.3 times the van der Waals volume of the constituent monomers (75). The van der Waals volumes of monomer units are usually estimated using Bondi's group contribution method (76). As permeability depends on both solubility and diffusivity, one may consider the effect of free volume on solubility and diffusion coefficients individually when assessing the impact of free volume on permeation properties. As discussed in more detail below, the effect of chain packing on gas diffusion coefficients in amorphous glassy polymers is stronger than the effect of chain packing on gas solubility coefficients. Thus, correlations of permeability with free volume often closely resemble correlations of diffusion coefficients with free volume. In polymers with highly ordered regions, such as semicrystalline or liquid crystalline polymers, the effect of free volume on solubility is stronger than in amorphous materials. In ordered or partially ordered polymers, solubility is sensitive to the amount of ordered material and the efficiency of packing in the ordered regions (6, 70, 77). The effect of free volume on penetrant diffusion coefficients in polymers is often described using concepts from the Cohen and Turnbull model (18). This statistical mechanics model provides a simplistic description of diffusion in a liquid of hard spheres. A hard sphere penetrant is considered to be trapped in a virtual cage created by its neighbors. Free volume is defined as the volume of the cage less the volume of the penetrant. Free volume fluctuations, which occur randomly due to thermallystimulated Brownian motion of neighboring hard spheres, provide opportunities for the penetrant to execute a diffusion step if the gap (Le. free volume fluctuation) occurs sufficiently close to the penetrant to be accessible and is of sufficient size to accommodate it. The diffusion coefficient of a penetrant is given by:


0

(4) where ν is the size of afreevolume element, v* is the minimum free volume element size which can accommodate the penetrant, F(v) is the contribution of free volume elements of size ν to the total diffusion coefficient, and p(v) is the probability of finding a free volume element of size between ν and v+dv. The distribution of free volume is obtained, using standard techniques of statistical mechanics, by maximizing, at fixed number of molecules and fixed total free volume, the excess entropy resulting from the distribution of free volume. The result is:

where γ is an overlap parameter introduced to avoid double counting of free volume elements shared by more than one hard sphere, and is the average free volume.



309 As average free volume in a polymer matrix cannot be measured directly, in Equation 5 is usually replaced by fractional free volume, which is calculated based on polymer density and an estimate of the occupied volume in the polymer. Free volume distributions calculated using Equation 5 are presented in Figure 1 for several values of average free volume. From Figure 1, this theory predicts that smaller free volume elements are more numerous than larger ones. Qualitatively, this prediction is consistent with molecular simulations of free volume distributions in glassy polymers such as atactic poly(propylene)(79) and poly(vinylchloride) (20). As indicated in Figure 1, the theory also predicts that the probability of finding a free volume element of a particular size increases as average free volume increases. Using these results, the Cohen and Turnbull model provides the following expression for the penetrant diffusion coefficient (18): (6) where A is a pre-exponential factor which depends weakly on temperature. Penetrant size is strongly correlated with v*. From Equation 6, penetrant diffusion coefficients decrease exponentially with increasing penetrant size (v*), which is qualitatively consistent with experimental observations for infinite dilution penetrant diffusion coefficients in glassy polymers (14). This model, based on diffusion of spheres, cannot provide insight into the effect of penetrant shape on diffusion coefficients. Equation 6 also predicts that, in two polymer matrices with very different average free volume, the effect of penetrant size on diffusion coefficients is weaker in the higher free volume polymer. While this model is obviously a crude approximation of the complex cooperative segmental chain dynamics which govern penetrant transport in polymers, it provides an intuitively useful qualitative rationale for the effect of free volume and penetrant size on transport properties of polymers. Low Permeability Liquid Crystalline Polymers Liquid crystalline polymers such as PPTA and HBA/HNA copolymers have remarkably high barrier properties (6,10,11, 21-23). Values of oxygen permeability. for several glassy LCPs and for conventional, non-liquid crystalline high barrier glassy polymers are presented in Table 1. Oxygen permeability is at least an order of magnitude lower in LCPs than in common glassy polymers such as PET and PVC and can be comparable to oxygen permeability in PAN, a noted barrier polymer. The influence of chain packing (i.e. free volume) on solubility, diffusivity and permeability in liquid crystalline polymers can be studied by comparing properties of LCPs in the disordered, isotropic state with those in the ordered, liquid crystalline state. HIQ-40 is a random, glassy, thermotropic, nematogenic terpolymer synthesized from 40 mole percent p-hydroxybenzoic acid and 30 mole percent each of isophthalic acid and hydroquinone. The chemical structures of the constituent monomers for HIQ-40 are:

p-hydroxybenzoic acid

isophthalic acid

hydroquinone


310


HIQ-40 may be dissolved in volatile solvents and cast into thin, metastable, glassy, optically isotropic, transparent films at ambient conditions. The rapid evaporation of volatile solvent molecules, in comparison with the characteristic rate of mesogen ordering, results in the preparation of solvent-free, isotropic, and disordered amorphous films which pack as illustrated in the cartoon of Figure 2a. Subsequent heating in the range of the glass transition temperature, Tg, confers sufficient chain mobility to permit the development of axial (Le. nematic liquid crystalline) order and small amounts ( ο α

σ

χ

150 h

ο


"Ε Ο Û. Ι Ο)

Ο*

100 h

3

CO

0 '—'— — — —'— — — — — — — — —' ο 1

0

50

1

1

100

1

1

1

150 200 T[°C]

1

1

250

1

1

300

1

350

Figure 3. Effect of annealing temperature on acetone solubility (open circles) and diffusivity (filled circles) in HIQ-40 samples at 35°C and an acetone relative pressure of 0.15. 1000

ι

100

D X

Ο

3

Figure 4. Correlation of acetone solubility and diffusivity in HIQ-40 with fractional free volume (6, 23). The solubility and diffusivity were determined at 35°C and an acetone relative pressure of 0.15.


314 If the ordering of the liquid crystalline phase precludes solubility within the HIQ40 domains just as conventional crystallinity typically precludes sorption and transport in the crystallites, then penetrant molecules would be largely restricted to boundary regions between domains. Since the domain boundaries account for a small fraction of the sample mass (28), penetrant solubilities based upon the overall sample weight would appear to be quite low relative to solubilities in conventional amorphous polymers. Moreover, the domains could act as impenetrable barriers to diffusion of small molecules, providing a tortuous path for the penetrant molecules. In HBA/HNA73/27, gas and vapor solubility and transport data are consistent with models suggesting that only domain boundary regions separating nematic liquid crystalline domains are accessible for penetrant sorption and transport (8 22). Downloaded by UNIV OF QUEENSLAND on October 6, 2015 | http://pubs.acs.org Publication Date: January 28, 1999 | doi: 10.1021/bk-1998-0710.ch021

f

Permeability of Gases and Acetone in HIQ-40. Figure 5 presents the relative permeability of HIQ-40 to a series of gases as a function of annealing temperature. The permeability of annealed samples are reported relative to the permeability of an unannealed (Le. as-cast) sample. Acetone permeability is also presented in this figure and is calculated as the product of acetone solubility and diffusivity according to Equation 2. The numbers in parentheses are the kinetic diameters of the penetrant molecules. Kinetic diameter is a common measure of penetrant size. As shown in Figure 5, the permeability of all penetrants decreases with increasing annealing temperature up to 300°C. The larger penetrants are more strongly affected by annealing-induced ordering than the smaller penetrants. The results of the simple free volume theory of diffusion (Equation 6) suggest that since the larger penetrants require more free volume to execute a diffusion step, diffusion coefficients of larger penetrants should be more strongly influenced by a reduction in free volume than the diffusion coefficients of small penetrant molecules. Positron Annihilation Lifetime Spectroscopy of HIQ-40 Films. Positron annihilation lifetime spectroscopy has emerged as a sensitive technique to probe free volume in polymers (33, 34). PALS uses orthoPositronium [oPs] as a probe of free volume in the polymer matrix. oPs resides in regions of reduced electron density, such as free volume elements between and along chains and at chain ends (33). The lifetime of oPs in a polymer matrix, 13, reflects the mean size of free volume elements accessible to the oPs. The intensity of oPs annihilations in a polymer sample, I3, reflects the concentration of free volume elements accessible to oPs. The oPs lifetime in a polymer sample is finite (on the order of several nanoseconds), so PALS probes the accessibility of free volume elements on nanosecond timescales (35). Table Π presents PALS results and other physical property data for an as-cast HIQ-40 sample and for a sample that was annealed for one hour at 200°C. The annealing protocol results in a 2.5% increase in density, which corresponds to a 17% decrease in fractional free volume. The acetone diffusion coefficient decreases almost five-fold and acetone solubility decreases by approximately 90% as a result of the ordering induced by the annealing protocol. The oPs lifetime decreases by 14%, suggesting that the average free volume cavity size decreases due to annealing. Based on the oPs lifetime, the mean free volume cavity diameter may be estimated (36); these values are reported in parentheses in Table II. The PALS I3 parameter, which reflects the relative concentration of free volume elements in the polymer matrix, is approximately 22% lower in the annealed, liquid crystalline sample. The free volume accessible to oPs is more than 50% lower in the annealed sample. This decrease is much larger than the decrease in free volume probed by density, and suggests that both the amount of free volume and the accessibility of free volume over the lifetime of the oPs probe decrease as the isotropic as-cast sample orders towards the liquid crystalline state. This finding is consistent with the much



315


316 higher glass transition temperature, in the annealed, nematic, liquid crystalline state than in the as-cast, isotropic state. The large increase in T upon annealing suggests rather profound decreases in segmental mobility upon ordering the polymer from the isotropic to the liquid crystalline state. These composite results indicate that liquid crystalline ordering in aromatic polyesters such as HIQ-40 can strongly decrease penetrant solubility and diffusivity while improving static packing efficiency, as characterized by density and FFV. Transport properties of larger penetrant molecules are more strongly affected by this ordering process than small penetrant molecules. The improvement in packing efficiency is accompanied by sharp decreases in segmental motions important in the glass-rubber transition and perhaps by decreases in polymer chain motion important for oPs accessibility in the polymer matrix. Downloaded by UNIV OF QUEENSLAND on October 6, 2015 | http://pubs.acs.org Publication Date: January 28, 1999 | doi: 10.1021/bk-1998-0710.ch021

g

Table Π. Physical properties and PALS results for HIQ-40 films Property Density [gm/cm ] ±0.001 (6) FFV (6) T [K] (57) Relative Free Volume Cavity Size, 3

g

i [ n s ] ± 0.03(57) Relative Free Volume Concentration, I [ % ] ± 0.3(57) Relative Free Volume Accessible to Positrons, 3

As-Cast 1.374 0.128 315 1.80 (5.3Â) 8

Annealed % Change +2.5 1.408 0.106 -17 412 +31 1.54 -14 (4.7Â)»

15.2

11.8

-22

89.4

42.9

-52

3

3

t I 3

b 3

3

[ns %] (57) 3

2

20

220

3

Acetone Solubility [cm (STP)/(cm atm)] (6) Acetone Diffusivity [cm /s] (6)

48xl0-

13

lOx 10-

-87 13

-79

a

Mean free volume cavity diameter calculated according to the free volume model of Jean (56, 58). Based on a model of spherical free volume cavities, the free volume accessible to oPs may be estimated from the product of the cube of the cavity size, 13, and the concentration of free volume cavities, I3 (55). b

Intermediate Permeability Amorphous Polymers In contrast to the LCP results just presented, in glassy polymers used as gas separation membranes, free volume influences diffusion coefficients much more than solubility coefficients. Figure 6 provides an example of this effect. In thisfigure,the solubility, diffusivity, and permeability of methane in a series of glassy, aromatic, amorphous poly(isophthalamides) [PIPAs] are presented as a function of the fractional free volume in the polymer matrix. (More complete descriptions of the transport properties of this family of materials are available elsewhere (59, 40)). The fractional free volume is manipulated systematically in this family of glassy polymers by synthesizing polymers with different substituent and backbone elements as shown in



317

Ο τ-

Ο

Ο)

χ ε ο

ε ο

φ

ε

ε Φ Χ

ο

ο

CL Ι Ο) w

ε υ 10

11

Figure 6. Methane solubility, diffusivity, and permeability of a series of low free volume polyisophthalamides. The sorption and transport data were determined at 35°C and an upstream pressure of 3 atmospheres.


318


Table ΠΙ. From Figure 6a, there is no systematic trend of gas solubility with fractional free volume in these materials. Other studies have found a small decrease in gas solubility with decreasing free volume (14). The interchain gaps which are the locus of sorption in polymers are probably more accessible to penetrants in amorphous polymers than in crystalline or liquid crystalline regions, where these gaps may be absent or so small as to be inaccessible or the gaps may be locked into rigid, ordered regions where the chain packing is too energetically difficult to disturb to permit dissolution of penetrant molecules. The tendency of a penetrant molecule to dissolve in an amorphous polymer matrix depends strongly on penetrant condensability and the strength of specific polymer-penetrant interactions and weakly on the free volume of the polymer matrix (14). Table III. Chemical structure, physical properties, and PALS results for poly(isophthalamides) presented in Figure 6. x

:

Symbol

V - '•.

R

a b c d

H C H

S0

e f g h

6

SO2

5

C(CH ) Si(CH ) H 3

SO2

3

3

C H C(CH ) Si(CH ) 6

2

3

5

3

3

3

3

SO2

C(CF ) C(CF ) C(CF ) C(CF ) 3

2

3

2

3

2

3

2

Tg[°C]

FFV

323 328 337 273 297 311 315 272

0.100 0.107 0.110 0.123 0.149 0.152 0.153 0.156

τ [ns] 1.60 1.69 1.81

l3t%] 18.3 17.4 19.6

2.11 2.24 2.55 2.64

20.7 20.2 20.3 20.5

3

τ

3

3

3

[ns ] 4.07 4.84 5.90 9.35 11.3 16.6 18.5

X Figures 6b and 6c present the effect of free volume on methane diffusivity and permeability, respectively. Over the range of free volume explored (0.1 in PTFE, only 0.4% in TFE/PDD 65 and TFE/PDD87 and only 0.2% in PTMSP. Thus, the vast majority of the free volume accessible to oPs is in the larger free volume elements. For the TFE/PDD copolymers, x is about 20% larger in TFE/PDD87 than in TFE/PDD65 suggesting that the larger free volume elements accessible to oPs have an average volume ( τ ) approximately 100% larger in TFE/PDD87 than in TFE/PDD65. τ is substantially higher in PTMSP than in TFE/PDD65, TFE/PDD87 or PTFE, suggesting that the largest free volume elements accessible to oPs in PTMSP are significandy larger than the largest free volume elements in the TFE/PDD copolymers or PTFE. The relative concentration of the free volume elements, I and I4, is 20% and 10% lower, respectively, in TFE/PDD87 than in TFE/PDD65. Thus, while the size of these elements is higher in the copolymer with more PDD, the concentration is lower. The net relative free volume accessible to oPs (T I +X4 l4> is 80% larger in TFE/PDD87 than in TFE/PDD65. TFE/PDD65 has been examined using PALS by Davies and Pethrick (5 Q. Whilst these authors found two oPs localization sites in the polymer, they did not comment on the possible free volume structure responsible for these two oPs components. The size of the larger cavities was postulated to be the controlling factor in the high gas diffusion rates of TFE/PDD65. The size and concentration of the large free volume elements, and consequently the overall free volume, available for penetrant transport are much larger in the TFE/PDD copolymers than in PTFE. This result is consistent with the notion that the addition of PDD to TFE frustrates chain packing, rendering the resulting copolymers totally amorphous and dramatically more permeable than PTFE. PTMSP has the largest 14 and I4 of all of the polymers considered and, as a result, has an enormously higher accessible free volume to oPs than the other materials. This result is consistent with the oxygen permeability data, where PTMSP is more than seven times as permeable to oxygen as TFE/PDD87, the most permeable TFE/PDD copolymer. These results suggest that the free volume accessible to oPs in the copolymers is much lower than that in PTMSP, which is intriguing since TFE/PDD87 and PTMSP have essentially the same fractional free volume (as characterized by density and group contribution estimates of occupied volume). Relative to the copolymers, PTMSP has a much higher concentration of both large and small free volume elements accessible to oPs over the nanosecond time scale of the experiment, and the largest free volume elements in PTMSP are substantially larger than free volume elements in the TFE/PDD copolymers. Most of the difference in relative free volume estimated by PALS is due to the contribution from the large free volume elements in PTMSP (4 $. In this regard, Consolati et φ $ and YampoFskii et a(4 Tf have attributed the smaller cavities in PTMSP to a channel structure and the larger cavities to conventional inter-and intra-chain free volume. These composite results suggest that the distribution and availability of free volume in PTMSP and the TFE/PDD copolymers are very different. Both PTMSP and the TFE/PDD copolymers are high T , stiff chain materials, so it is unlikely that the vast differences in accessible free volume and permeability coefficients is solely related to great differences in segmental dynamics between these materials which would render the free volume in PTMSP much more accessible on the time scales appropriate for PALS and permeation. Rather, it seems more likely that free volume elements in PTMSP are interconnected and span the sample, providing extremely efficient pathways for penetrant diffusion. In fact, the notion of interconnected free volume elements in PTMSP has been invoked to explain the unusual transport 3

3

3

4

3

4


4

3

3

3

3

3

g


323 properties in this material (49, 51, 52). In many respects, gas and vapor transport in PTMSP is more similar to transport in microporous carbon than to transport in conventional glassy polymeric gas separation membranes. In the TFE/PDD copolymers, the free volume elements may be much more finely dispersed than in PTMSP and are not interconnected on the timescale of the PALS measurement. In fact, we have observed that gas sorption into the nonequilibrium excess volume in TFE/PDD copolymers is only 25% of the value expected based on macroscopic dilatometry measurements, suggesting that a substantial amount of free volume in the TFE/PDD copolymer is inaccessible to even the smallest penetrants.(5J)


Conclusions The most permeable polymers are amorphous glassy polymers in which chain packing is sufficiently poor to permit penetrant access. These polymers are distinguished by their high values of free volume. Moreover, the distribution of free volume may be important to the permeability in such polymers. At the same average free volume, polymers with interconnected free volume elements may have extraordinarily high transport properties relative to polymers in which the free volume distribution does not favor permanently connected free volume elements. In this regard, PALS provides a very useful probe of free volume distribution in such materials and provides results which are more consistent with gas permeability results than density-based fractional free volume. The least permeable polymers are glassy polymers in which the backbone structure permits very efficient chain packing in the solid state. Liquid crystalline polymers provide one example of such classes of materials. As liquid crystalline order is perfected in the glassy copolyester considered and free volume is reduced in the polymer matrix, both penetrant diffusivity and solubility decrease strongly, and the permeability properties of larger penetrants are influenced to a larger extent than those of smaller penetrants. Systematic manipulation of free volume and free volume distribution via backbone structure and processing results in many orders of magnitude change in permeability properties. Acknowledgments The authors would like to acknowledge partial support of this work by the National Science Foundation (Young Investigator Award CTS-9257911-BDF). Literature Cited 1. 2. 3. 4. 5. 6. 7. 8.

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22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.

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325 37. 38. 39. 40.


41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53.

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