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Free volume model for molecular weights of polymers
Nuclear lnstnments &Methods in Physics Research Section B
Nuclear Instruments and Methods in Physics Research B63 (1992) 477-483
North-HolIand
Free volume model for molecular weights J.J. Singh a and A. Eftekhari
of polymers
b
” NASA Langley Research Center, Hampton, VA 23665, USA * Hampton Vniuersity, Hampton, VA 23668, USA
Received 6 August 1991 and in revised form 8 October 1991
A free volume model has been developed for determining molecular weights of linear polymers. It is based on the size of free volume cells in two geometries of poly (arylene ether ketonejs. Free volume cell sizes in test samples were measured using positron lifetime spectroscopy. The molecular weights computed from free volume cell sizes are in good agreement with the values measured by gel permeation chromatography, with a low angle laser light scattering photometer as the detector. The model has been further tested on two atactic polystyrene samples, where it predicted the ratio of their molecular weights with reasonable accuracy. 1. Introduction
All polymers generally have regions of low atomic density (free volume cells) resulting from geometrical mismatch among neighboring molecular chains/ segments. The sizes and concentrations of these free volume cells depend on the polymer molecular architecture and affect many physical properties of high polymers. For example, we have recently shown in ref. [l] that both the solubility coefficient as well as the diffusion coefficient for gaseous molecules are modified by the presence of free volume cells in the test polymers. As a result, the presence of free volume cells in the test polymers enhances the permeation of gases through them. Similarly, the free volume cells in polymeric materials affect their other thermodynamic properties. The free volume cell sizes change markedly at the phase transition temperatures of the test polymers. Thus, the large differences in thermal expansion coefficients, observed above and beIow the glass transition temperatures, arise from the large free volume expansion at the glass transition temperatures. The rapid decrease of free volume cell size with increase in pressure above the glass transition temperature also accounts for the observed trends in the compressibility of polymeric materials as a function of temperature. Positron lifetime spectroscopy provides direct information about the size and shape of free volume cells in polymeric materials. It thus enables separation of free volume compressibility from the compressibility of the excluded volume of the test polymers. Many conventional techniques can yield only a total compressibility that includes both effects. Excellent reviews of positron lifetime spectroscopy for microstructural analysis of polymers can be found in several recent reports [2,3]. 01~8-583X/92/~05.00
It is clear that the free volume plays an important role in determining physical properties of the polymeric mater&. In the case of linear polymers, a relationship between the free volume cell size (V,) and the molecular weight of the polymer (M) can be developed. Let the molecular weight of the polymer be M. For n moles of the polymer, ?zM=pV.
(1)
If the polymer has free volume, AV, we have:
=nM(l
+of),
(2)
where cy= (p’/p) (constant for a polymer system) and f = free volume fraction. Rearranging eq. (21, we have AMM/M=af.
(3)
The free volume fraction, f, can be written as [l)
f = cr,v,,
Pa)
where C is the polymer structural constant, Z, is the intensity of the third (longest) positron lifetime component in the test material and Vt is the average free volume cell size. It follows from eq. (3) $
= (aCI,)Vt.
(4)
It thus appears that the free volume cell size (V,> is a function of M. The relationship between V, and M can be expressed as follows ‘I. Vf =AMB. a’ Preliminary
(5) results based on this formalism have been
reported previously in ref. [4].
0 1992 - Elsevier Science Publishers B.V. All rights reserved
J.J. Singh, A. Eftekhari / Molecularweightsof polymers
418
A and B are constants for the test polymer and are expected to be independent of its degree of polymerization. However, they depend on the temperature and pressure of the sample environment. Eq. (5) is quite similar in form to the Mark-Houwink equation relating the intrinsic viscosity (7) and the molecular weight 04) of the polymer in solution [5]. It is the purpose of this study to make a systematic jnvestigation of V, vs 1M in selected linear polymers. Poly arylene ether ketones (PAEK) of variable molecular weights were selected as the test polymers. V, values were measured using positron lifetime spectroscopy (PLS). All measurements were made at room temperature and atmospheric pressure.
Fig. 1. Structure of (a) Meta and (b) Para polyarylene ether ketones.
were cooled rapidly to room temperature by removing the plates from the oven and placing them on a cool surface. Wide angle X-ray scattering showed the films to be free of detectable crystallinity.
2. Experimental procedure 2.2. Positron lifetime measurements 2.1. Sample preparation The structures of poly (arylene ether ketonels chosen for study are shown in fig. 1. It was anticipated that the different geometries of the two repeat units might lead to different packing behavior and hence different free volumes. Details of their syntheses and fractionation have been described elsewhere [6]. Physical properties of different fractions are summarized in table 1. Molecular weights of the various fractions in the two geometries differed by about a factor of 3, although the specific volumes of both polymer systems were practically independent of molecular weights in this range. Thin (30 urn) films, for the positron lifetime studies, were prepared by casting on glass plates from m-cresol solution. These films were air-dried for two days in a low-humidity enclosure, then gradually heated in an air-circulating oven to 25o”C, which is well above the polymer glass transition temperature, TB. Finally, they
In the case of thin PAEK films, positron lifetime measurements were made using a recently developed low energy positron flux generator [7]. Briefly, a 250 PC 22Na source, deposited on a 2.5 urn thick aluminized mylar film, is sandwiched between two 0.0127 cm thick 2.54 x 2.54 cm* tungsten strips. Two 2.54 X 2.54 cm’ test PAEK films insulate the two tungsten moderator strips from the aluminized mylar source holder. A potential difference of about 60 V is applied between the tungsten strips and the source holder. Thermalized positrons, diffusing out of the moderator strips, are attracted to the source holder when it is at a negative potential. These positrons have to drift through the PAEK films in order to reach the source holder. They will annihilate in the PAEK films before reaching the source holder. On the other hand, the leaking positrons are forced back into the moderator strips when the source holder is at a positive potential.
Table 1 Characteristics of polymer fraction [61 Sample
P
Mol. weight [10V3 Ml M, a)
M, b,
MW
[g/cm31
Para: Whole polymer Fraction A Fraction B, Fraction C Fraction D
_ 29.6 28.4 15.4 17.8
19.0 47.0 44.5 17.6 16.3
26.0 51.0 45.9 18.3 16.9
1.2058 1.2060 1.2041 1.2052 1.2061
0.50 0.78 0.70 0.37 0.38
16.5 163 159 160
Meta: Whole polymer Fraction AB, Fraction C
47.3 20.3
20.8 60.8 19.8
26.6 66.0 21.0
1.2072 1.2077 1.2074
0.39 0.65 0.35
148 153 148
a) End group analysis. b, Gel permeation chromatography/low
angle laser light scattering technique; these values have an error of +_10%.
J.J. Singh,
A. Eftekhari
/ Molecular
419
of polymers
Source
0
4
weights
bias -60
volts -
10
Source
f
positron annihilation events
bias +60 volts
3
10
_..J
120
140
160
160
Channel
200
220
Number
240
260
(1 Channel=645
280
300
320
300
320
psec)
positron annihilation 3 events $0
10 120
140
160
180
Channel
200
220
Number
240
260
(1 Channel=64.5
280 psec)
Fig. 2. (a) Comparison between negative and positive source bias lifetime spectra in a Meta-PAEK film. (b) Positron lifetime spectrum in a Meta-PAEK film obtained by subtracting normalize positive bias spectrum from negative bias spectrum.
Thus more positrons annihilate in the tungsten moderator strips when the source potential is positive than when it is negative. Careful measurements with wellcharacterized Teflon films have shown that the excess of positrons stopping in the tungsten moderators when the source foil is at a positive potential is 4%. Thus, subtraction of 96% of the positive source bias lifetime spectrum from the negative source bias lifetime spectrum should give lifetime spectrum exclusively due to
positrons annihilating in the PAEK films. Positron lifetime measurements in thick (5 mm) polystyrene samples were made in conventional manner, using a 50 KC “Na source. In both cases, positron annihilation events were counted using the fast-fast coincidence timing technique. The coincidence system time resolution was of the order of 225 ps. All measurements were made at room temperature and atmospheric pressure. Typically, spectrum accumulation time was 4 hours and
J.J. Sing& A. Eftekhari / Moiecuiarweightsof polymers
480
L
Positron Annihilation events
Ii F
/ lo”“‘-
120
140
160
180
Channel
200
220
Number
240
260
280
(1 Channeg64.5
300
320
340
psec.)
Fig. 3. Positron lifetime spectrum in a Para-PAEK film obtained by subtracting normalized positive bias spectrum from negative bias spectrum.
it provided approximately lo6 counts in each spectrum. Fig. 2(a) shows a typical lifetime spectrum observed in Meta-PAEK films with &60 V potential difference between the aluminized source foil and the tungsten moderator strips. Fig. 2(b) shows the lifetime spectrum obtained by subtracting normalized positive source bias spectrum from the negative source bias spectrum. This spectrum thus represents the lifetime spectrum of positrons annihilating in the Meta-PAEK films only. Fig. 3 shows a typical positron lifetime spectrum in Para-PAEK films. The lifetime spectra were analyzed using PAPLS [S] and POSITRONFIT EXTENDED [9] computer programs. In all cases, 3-component analyses gave the best fits to the experimental data.
3. Experimental results The positron lifetime results in Para-PAEK films are summarized in table 2. pi, Z, and r2, Z2 refer to the lifetime and intensity of the first and second lifetime components, respectively. The largest component lifetime (r3) results from orthopositronium pick-off annihilation. The lifetime (r3) is related to the free volume cell size (V,) as follows [lo]: 1 -=I--Rcsin27rz, 273
RO
1
R 0
where r3 = long component lifetime in ns, R = free volume cell radius in nm and R, = (R + AR) nm (the best fitted value of AR for all known data has been reported to be 0.1659 nm). The free cell volume (V,) is given by $rR3. The V, values for various Para-PAEK samples are summarized
in table 3. The positron lifetime results in Meta-PAEK films are summarized in table 4. The corresponding V, values are summarized in table 5. 3.1. Calculation of the molecular weights In order to obtain the values of A and B in eq. (51, experimental values of the test film molecular weights 04,) determined by the Gel Permeation Chromatography/ Low Angle Laser Light Scattering (GPC/ LALLS) technique [6] were used. There were five Para-PAEK and three Meta-PAEK samples of different molecular weights. We used experimental molecular weights of three Para-PAEK samples to determine the constants A and B by least squares method. These constants were then used to predict the molecular weights of the remaining two samples. Similarly, two experimental molecular weights of the Meta-PAEK samples were used to determine the structural constant A. The constant B is expected to be the same as for Para-PAEK samples #2. The meta A and B combination of constants was then used to predict the molecular weight of the third meta sample. Comparisons between the experimental and calculated M,, values for the two systems are summarized in table 3 and 5, respectively. It is apparent that the agreement is quite good for both the polymer systems. This indicates that eq. (5) represents the correct form of relationship between V, and #2 By comparing separate least square fits of 5 Para-PAEK and 3 Meta-PAEK samples, it has been determined that the constant B is the same for two geometries of PAEK samples. The constant A, however, is different for the two geometries as expected.
481
J.J. Singh, A. Eftekhari / Molecular weights of polymers Table 2 Positron lifetime
parameters
in Para-poly
(arylene
ether
ketone)
films
Geometry
71 [PSI
I, [%I
72 [PSI
I, [%I
73
Whole polymer Fraction A Fraction B, Fraction C Fraction D
187k12 171+09 139+ 16 196 f 08 165 + 17
37+3 44+3 35*4 41k2 33+5
409+ 16 407,16 371 f 24 416+20 390 f 24
59+3 53+2 60+2 56+3 63+3
905 f 60 106f 70 1037f70 869 f 75 850 + 65
Table 3 Summary
of Vr and saturation
moisture
Sample
Whole polymer Fraction A ‘) Fraction B, d, Fraction C d, Fraction D ‘)
‘)
‘) M,, values determined by + 10%). b, M, values are calculated ‘) These three samples were d, The molecular weights of
Table 4 Positron lifetime
parameters
contents
(V/o)
in Para-poly
(arylene
Mol. weight (IV,) GPC/LALLS a) (experimental) x10-3
Vrt A31
19.0 47.0 44.5 17.6 16.3
13.7 + 0.6 23.2 i 0.7 21.3kO.6 12.1+ 0.8 11.2+0.9
gel permeation
chromatography/low
ether
in Meta-Poly
(arylene
ether
ketone)
I, [%I
T* [PSI
12
51+2 59,l
Fraction
181+7
46+3
442+ 8 516k23 433 f 12
43&-l 36+1 47+2
Whole polymer d, Fraction AB, ‘) Fraction C ‘)
1.10 2.17 1.42 1.12 1.13
21.3 + 1.5 51.3f2.4 43.2 k 1.9 17.5k1.9 15.5 f 2.0 technique
(all values have an error of
films
183+4 211+3
Sample
M.W. (IV,) calculated ‘) x10-3
by using the equation: V, = AM:, where A = 2.70~ lo-* and B = 0.625. used to calculate the constants A and E. these samples were calculated using the values of A and B computed above.
7, [PSI
of Vr and saturation
4+1 3*1 6kl 3fl 4+1
[%I
angle laser light scattering
Geometry
Table 5 Summary
13 [%I
films
V/o
Whole polymer Fraction AB, C
ketone)
[PSI
moisture
contents
(V/o)
in Meta-poly
(arylene
[%I
ether
73
Ij [%I
[PSI
1698 f 40 2273 + 77 1651 *I52
ketone)
6+1 5*1 7*1
films
Mol. weight CM,) GPC/LALLS (experimental) ‘) x10-3
Gi
v/o
(A”)
(o/o)
M.W. CM,) calculated b, x10-s
20.8 60.8 19.8
70.8kO.4 130.2 _+0.4 66.2+0.6
0.43 0.47 1.06
22.8 f 0.2 60.3 f 0.3 20.4 f 0.3
‘) M, values determined by gel permeation chromatography/low angle laser light scattering technique tall values have an error of + 10%). b, M, values are calculated by using the equation: Vr = AM:, where A = 1.340X 10-l and B = 0.625. ‘) These two samples were used to calculate the constant A. d, The molecular weight of this sample was calculated using the value of A computed above.
J.J. Singh, A. Eftekhari / Molecular weights of polymers
482
A
Bl
Whole
C
Fraction Fig. 4. Comparison between molecular weights determined by GPCfLALLS
M. The comparison between the molecular weights measured by GPC/LALLS technique and the free volume model is illustrated in fig. 4. Eq. (5) should be applicable for the entire range of orthopositronium lifetimes observed in linear polymers (namely, 0.7-4.8 ns). These lifetimes correspond to molecular weights ranging from 4 X 103-4 X 10” amu. 3.2. Test of the model The model was further tested on two atactic polystyrene samples having well defined physical properties. These samples were compression molded, under 20 tons of pressure, into 13 mm diameter x 5 mm thick discs at 393 K. The discs were allowed to cool down to room temperature before positron lifetime measurements were made in them. Using the positron lifetime data in the two samples, the ratio of their molecular weights was calculated and compared with the reported value [ll]. The results are summarized in table 6. The agreement is reasonably good.
Table 6 Comparison between the experimenatal Sample No.
1 2
M, (*5%)
13~~ 96000
and PLS methods.
4. Discussion Several authors have previously investigated the molecular weight dependence of positron decay rates in polymers. Chuang et al. 1121studied positron annihilation in polypropylene of various weight-averaged molecular weights and found a linear decrease in pickoff annihilation lifetime with increasing molecular weights. West et al. 1131,on the other hand, noted that pick-off annihilation lifetime in polystyrene samples is essentially independent of their molecular weights in the range 103-lo6 amu, in contrast to the situation in Teflon. We have conducted a study of pick-off annihilation lifetimes in amorphous linear polymers with molecular weights in range of 1.5 x 104-1.5 X lo5 amu. Instead of the pick-off annihilation lifetime, we have selected the average free volume cell size (V,) as the parameter of interest, since it is more sensitive to molecular weight variation than the lifetime. From the data described earlier, it appears that V, values correlate with the M, values of the test poly-
and reported values of the molecular weights of atactic polystyrene samples 73
[PSI
1980+ 5 1780 + 10
Ratio of Molecular Weights a)
V,(A3)
9.5.0* 0.5 76.9 + 0.9
a) Calculations of ratio of molecular weights: Vf = AMB and therefore
V
2
= (2)‘: 2
MI/M,
(!&)1/B
1.35-+0.14
1.41t0.04
so we get (F)1/B f2
= M,, where B = 0.625. M2
J.J. Singh, A. Eftekhari / Molecular weights of polymers Table 7 Summary of Mark-Houwink Sample
Para-PAEK Meta-PAEK
and free volume model constants
Mark-Houwink equation a)
Free volume model b,
K
a
A
B
1.20x10-” 9.00x 1o-4
0.60 0.60
2.70~ 1tX2 1.34x10-’
0.625 0.625
‘) Mark-Houwink equation: n = KM”. b, Free volume modeI: V, = AMB.
mers quite well. As expected, a unique set of A and B values predict the test polymer M, values as long as their molecular structure remains unchanged. A change in structure as one goes from Para-PAEK to MetaPAEK geometry, necessitates a different set of constants. A notable feature is the fact that V, values are much larger in the meta geometry than the para geometry. This, however, is quite consistent with the different packing behavior of the two geometries. A further corroboration of the validity of the free volume model was obtained by comparing the MarkHouwink equation [5] and the free volume model constants. The results are summarized in table 7. Clearly, the M-H constants and the free volume model constants exhibit similar behavior when the repeat unit geometries change.
5. Concluding remarks It has been shown that positron lifetime spectroscopy may provide a reasonably accurate technique for measuring molecular weights of amorphous linear polymers. The proposed technique relates the free volume cell size (I’,) with the sample molecular weight (hl) in a manner remarkably similar to that obtained by Mark-Houwink (M-H) between the inherent viscosity (7) and molecular weight (M) of the polymer in solution. However, the M-H technique requires the sample to be in liquid form. The PLS technique, on the other hand, is applicable to samples in solid phase. It has been verified for two geometries of the PAEK where it predicts, with reasonable accuracy, not only the higher l$ values (i.e. looser packing) for the meta
483
geometry, but also the anticipated structural differences between the two geometries, The technique has been further tested on two polystyrene samples where it predicts the ratio of their molecular weights with reasonable accuracy.
Acknowledgements
The authors are grateful to Dr. Jeff A. Hinkley of Langley Research Center for preparing PAEK films and providing information about their physical properties and to Professor Phillips L. Jones of Duke University for supplying polystyrene samples used in this study.
References [II J.J. Singh, A. Eftekhari, B.T. Upchurch and KS. Burns, in: Proc. 3rd Int. Conf. on Positron and Positronium Chemistry, ed. Y.C. Jean (World Scientific, Singapore, 1990) p. 54. 121 Y.C. Jean, Microchem. Journal, 42 (1990) 72. 131Y.C. Jean, in: Proc. 3rd Int. Conf. on Positron and Positronium Chemistry, ed. Y.C. Jean (World Scientific, Singapore, 1990) p. 1. [41 J.J. Singh, A. Eftekhari and T.L. St. Clair, A Study of Physical Properties of ODPA-p-PDA Polyimide Films, NASA Technical Memorandum 102625 (1990). 151 R.B. Seymour and C.E. Carraher, Structure-Property Relationships in Polymers (Plenum, New York, 1984) p. 104. [61 J.A. Hinkley, K.A. Crook and J.R.J. Davis, High Performance Polymers 1 (1988) 61. [71 J.J. Singh, A. Eftekhari and T.L. St. Clair, Nucl. Instr. and Meth. B53 (1991) 342. [81 J.J. Singh, G.H. Mall and D.R. Sprinkle, Analysis of Positron Lifetime Spectra in Polymers, NASA TP-2853 (1988). [91 P. Kirkegaard, Comput. Phys. Commun. 7 (1974) 401. m H. Nakanishi, S.J. Wang, and Y.C. Jean, in: Positron Annihilation in Fluids, ed. S.C. Sharma (World Scientific, Singapore, 1988) p. 292. [111 A.D. Kasbekar, P.L. Jones and A. Crowson, J. Polym. Sci., Polym. Chem. 27 (1989) 1373. ml S.Y. Chuang and S.T. Tao, J. Appl. Phys. 44 (1973) 5171. [I31 D.H.D. West, V.J. McBrierty and C.F.G. Delany, Appl. Phys. (1975) 171.
Nuclear Instruments and Methods in Physics Research B63 (1992) 477-483
North-HolIand
Free volume model for molecular weights J.J. Singh a and A. Eftekhari
of polymers
b
” NASA Langley Research Center, Hampton, VA 23665, USA * Hampton Vniuersity, Hampton, VA 23668, USA
Received 6 August 1991 and in revised form 8 October 1991
A free volume model has been developed for determining molecular weights of linear polymers. It is based on the size of free volume cells in two geometries of poly (arylene ether ketonejs. Free volume cell sizes in test samples were measured using positron lifetime spectroscopy. The molecular weights computed from free volume cell sizes are in good agreement with the values measured by gel permeation chromatography, with a low angle laser light scattering photometer as the detector. The model has been further tested on two atactic polystyrene samples, where it predicted the ratio of their molecular weights with reasonable accuracy. 1. Introduction
All polymers generally have regions of low atomic density (free volume cells) resulting from geometrical mismatch among neighboring molecular chains/ segments. The sizes and concentrations of these free volume cells depend on the polymer molecular architecture and affect many physical properties of high polymers. For example, we have recently shown in ref. [l] that both the solubility coefficient as well as the diffusion coefficient for gaseous molecules are modified by the presence of free volume cells in the test polymers. As a result, the presence of free volume cells in the test polymers enhances the permeation of gases through them. Similarly, the free volume cells in polymeric materials affect their other thermodynamic properties. The free volume cell sizes change markedly at the phase transition temperatures of the test polymers. Thus, the large differences in thermal expansion coefficients, observed above and beIow the glass transition temperatures, arise from the large free volume expansion at the glass transition temperatures. The rapid decrease of free volume cell size with increase in pressure above the glass transition temperature also accounts for the observed trends in the compressibility of polymeric materials as a function of temperature. Positron lifetime spectroscopy provides direct information about the size and shape of free volume cells in polymeric materials. It thus enables separation of free volume compressibility from the compressibility of the excluded volume of the test polymers. Many conventional techniques can yield only a total compressibility that includes both effects. Excellent reviews of positron lifetime spectroscopy for microstructural analysis of polymers can be found in several recent reports [2,3]. 01~8-583X/92/~05.00
It is clear that the free volume plays an important role in determining physical properties of the polymeric mater&. In the case of linear polymers, a relationship between the free volume cell size (V,) and the molecular weight of the polymer (M) can be developed. Let the molecular weight of the polymer be M. For n moles of the polymer, ?zM=pV.
(1)
If the polymer has free volume, AV, we have:
=nM(l
+of),
(2)
where cy= (p’/p) (constant for a polymer system) and f = free volume fraction. Rearranging eq. (21, we have AMM/M=af.
(3)
The free volume fraction, f, can be written as [l)
f = cr,v,,
Pa)
where C is the polymer structural constant, Z, is the intensity of the third (longest) positron lifetime component in the test material and Vt is the average free volume cell size. It follows from eq. (3) $
= (aCI,)Vt.
(4)
It thus appears that the free volume cell size (V,> is a function of M. The relationship between V, and M can be expressed as follows ‘I. Vf =AMB. a’ Preliminary
(5) results based on this formalism have been
reported previously in ref. [4].
0 1992 - Elsevier Science Publishers B.V. All rights reserved
J.J. Singh, A. Eftekhari / Molecularweightsof polymers
418
A and B are constants for the test polymer and are expected to be independent of its degree of polymerization. However, they depend on the temperature and pressure of the sample environment. Eq. (5) is quite similar in form to the Mark-Houwink equation relating the intrinsic viscosity (7) and the molecular weight 04) of the polymer in solution [5]. It is the purpose of this study to make a systematic jnvestigation of V, vs 1M in selected linear polymers. Poly arylene ether ketones (PAEK) of variable molecular weights were selected as the test polymers. V, values were measured using positron lifetime spectroscopy (PLS). All measurements were made at room temperature and atmospheric pressure.
Fig. 1. Structure of (a) Meta and (b) Para polyarylene ether ketones.
were cooled rapidly to room temperature by removing the plates from the oven and placing them on a cool surface. Wide angle X-ray scattering showed the films to be free of detectable crystallinity.
2. Experimental procedure 2.2. Positron lifetime measurements 2.1. Sample preparation The structures of poly (arylene ether ketonels chosen for study are shown in fig. 1. It was anticipated that the different geometries of the two repeat units might lead to different packing behavior and hence different free volumes. Details of their syntheses and fractionation have been described elsewhere [6]. Physical properties of different fractions are summarized in table 1. Molecular weights of the various fractions in the two geometries differed by about a factor of 3, although the specific volumes of both polymer systems were practically independent of molecular weights in this range. Thin (30 urn) films, for the positron lifetime studies, were prepared by casting on glass plates from m-cresol solution. These films were air-dried for two days in a low-humidity enclosure, then gradually heated in an air-circulating oven to 25o”C, which is well above the polymer glass transition temperature, TB. Finally, they
In the case of thin PAEK films, positron lifetime measurements were made using a recently developed low energy positron flux generator [7]. Briefly, a 250 PC 22Na source, deposited on a 2.5 urn thick aluminized mylar film, is sandwiched between two 0.0127 cm thick 2.54 x 2.54 cm* tungsten strips. Two 2.54 X 2.54 cm’ test PAEK films insulate the two tungsten moderator strips from the aluminized mylar source holder. A potential difference of about 60 V is applied between the tungsten strips and the source holder. Thermalized positrons, diffusing out of the moderator strips, are attracted to the source holder when it is at a negative potential. These positrons have to drift through the PAEK films in order to reach the source holder. They will annihilate in the PAEK films before reaching the source holder. On the other hand, the leaking positrons are forced back into the moderator strips when the source holder is at a positive potential.
Table 1 Characteristics of polymer fraction [61 Sample
P
Mol. weight [10V3 Ml M, a)
M, b,
MW
[g/cm31
Para: Whole polymer Fraction A Fraction B, Fraction C Fraction D
_ 29.6 28.4 15.4 17.8
19.0 47.0 44.5 17.6 16.3
26.0 51.0 45.9 18.3 16.9
1.2058 1.2060 1.2041 1.2052 1.2061
0.50 0.78 0.70 0.37 0.38
16.5 163 159 160
Meta: Whole polymer Fraction AB, Fraction C
47.3 20.3
20.8 60.8 19.8
26.6 66.0 21.0
1.2072 1.2077 1.2074
0.39 0.65 0.35
148 153 148
a) End group analysis. b, Gel permeation chromatography/low
angle laser light scattering technique; these values have an error of +_10%.
J.J. Singh,
A. Eftekhari
/ Molecular
419
of polymers
Source
0
4
weights
bias -60
volts -
10
Source
f
positron annihilation events
bias +60 volts
3
10
_..J
120
140
160
160
Channel
200
220
Number
240
260
(1 Channel=645
280
300
320
300
320
psec)
positron annihilation 3 events $0
10 120
140
160
180
Channel
200
220
Number
240
260
(1 Channel=64.5
280 psec)
Fig. 2. (a) Comparison between negative and positive source bias lifetime spectra in a Meta-PAEK film. (b) Positron lifetime spectrum in a Meta-PAEK film obtained by subtracting normalize positive bias spectrum from negative bias spectrum.
Thus more positrons annihilate in the tungsten moderator strips when the source potential is positive than when it is negative. Careful measurements with wellcharacterized Teflon films have shown that the excess of positrons stopping in the tungsten moderators when the source foil is at a positive potential is 4%. Thus, subtraction of 96% of the positive source bias lifetime spectrum from the negative source bias lifetime spectrum should give lifetime spectrum exclusively due to
positrons annihilating in the PAEK films. Positron lifetime measurements in thick (5 mm) polystyrene samples were made in conventional manner, using a 50 KC “Na source. In both cases, positron annihilation events were counted using the fast-fast coincidence timing technique. The coincidence system time resolution was of the order of 225 ps. All measurements were made at room temperature and atmospheric pressure. Typically, spectrum accumulation time was 4 hours and
J.J. Sing& A. Eftekhari / Moiecuiarweightsof polymers
480
L
Positron Annihilation events
Ii F
/ lo”“‘-
120
140
160
180
Channel
200
220
Number
240
260
280
(1 Channeg64.5
300
320
340
psec.)
Fig. 3. Positron lifetime spectrum in a Para-PAEK film obtained by subtracting normalized positive bias spectrum from negative bias spectrum.
it provided approximately lo6 counts in each spectrum. Fig. 2(a) shows a typical lifetime spectrum observed in Meta-PAEK films with &60 V potential difference between the aluminized source foil and the tungsten moderator strips. Fig. 2(b) shows the lifetime spectrum obtained by subtracting normalized positive source bias spectrum from the negative source bias spectrum. This spectrum thus represents the lifetime spectrum of positrons annihilating in the Meta-PAEK films only. Fig. 3 shows a typical positron lifetime spectrum in Para-PAEK films. The lifetime spectra were analyzed using PAPLS [S] and POSITRONFIT EXTENDED [9] computer programs. In all cases, 3-component analyses gave the best fits to the experimental data.
3. Experimental results The positron lifetime results in Para-PAEK films are summarized in table 2. pi, Z, and r2, Z2 refer to the lifetime and intensity of the first and second lifetime components, respectively. The largest component lifetime (r3) results from orthopositronium pick-off annihilation. The lifetime (r3) is related to the free volume cell size (V,) as follows [lo]: 1 -=I--Rcsin27rz, 273
RO
1
R 0
where r3 = long component lifetime in ns, R = free volume cell radius in nm and R, = (R + AR) nm (the best fitted value of AR for all known data has been reported to be 0.1659 nm). The free cell volume (V,) is given by $rR3. The V, values for various Para-PAEK samples are summarized
in table 3. The positron lifetime results in Meta-PAEK films are summarized in table 4. The corresponding V, values are summarized in table 5. 3.1. Calculation of the molecular weights In order to obtain the values of A and B in eq. (51, experimental values of the test film molecular weights 04,) determined by the Gel Permeation Chromatography/ Low Angle Laser Light Scattering (GPC/ LALLS) technique [6] were used. There were five Para-PAEK and three Meta-PAEK samples of different molecular weights. We used experimental molecular weights of three Para-PAEK samples to determine the constants A and B by least squares method. These constants were then used to predict the molecular weights of the remaining two samples. Similarly, two experimental molecular weights of the Meta-PAEK samples were used to determine the structural constant A. The constant B is expected to be the same as for Para-PAEK samples #2. The meta A and B combination of constants was then used to predict the molecular weight of the third meta sample. Comparisons between the experimental and calculated M,, values for the two systems are summarized in table 3 and 5, respectively. It is apparent that the agreement is quite good for both the polymer systems. This indicates that eq. (5) represents the correct form of relationship between V, and #2 By comparing separate least square fits of 5 Para-PAEK and 3 Meta-PAEK samples, it has been determined that the constant B is the same for two geometries of PAEK samples. The constant A, however, is different for the two geometries as expected.
481
J.J. Singh, A. Eftekhari / Molecular weights of polymers Table 2 Positron lifetime
parameters
in Para-poly
(arylene
ether
ketone)
films
Geometry
71 [PSI
I, [%I
72 [PSI
I, [%I
73
Whole polymer Fraction A Fraction B, Fraction C Fraction D
187k12 171+09 139+ 16 196 f 08 165 + 17
37+3 44+3 35*4 41k2 33+5
409+ 16 407,16 371 f 24 416+20 390 f 24
59+3 53+2 60+2 56+3 63+3
905 f 60 106f 70 1037f70 869 f 75 850 + 65
Table 3 Summary
of Vr and saturation
moisture
Sample
Whole polymer Fraction A ‘) Fraction B, d, Fraction C d, Fraction D ‘)
‘)
‘) M,, values determined by + 10%). b, M, values are calculated ‘) These three samples were d, The molecular weights of
Table 4 Positron lifetime
parameters
contents
(V/o)
in Para-poly
(arylene
Mol. weight (IV,) GPC/LALLS a) (experimental) x10-3
Vrt A31
19.0 47.0 44.5 17.6 16.3
13.7 + 0.6 23.2 i 0.7 21.3kO.6 12.1+ 0.8 11.2+0.9
gel permeation
chromatography/low
ether
in Meta-Poly
(arylene
ether
ketone)
I, [%I
T* [PSI
12
51+2 59,l
Fraction
181+7
46+3
442+ 8 516k23 433 f 12
43&-l 36+1 47+2
Whole polymer d, Fraction AB, ‘) Fraction C ‘)
1.10 2.17 1.42 1.12 1.13
21.3 + 1.5 51.3f2.4 43.2 k 1.9 17.5k1.9 15.5 f 2.0 technique
(all values have an error of
films
183+4 211+3
Sample
M.W. (IV,) calculated ‘) x10-3
by using the equation: V, = AM:, where A = 2.70~ lo-* and B = 0.625. used to calculate the constants A and E. these samples were calculated using the values of A and B computed above.
7, [PSI
of Vr and saturation
4+1 3*1 6kl 3fl 4+1
[%I
angle laser light scattering
Geometry
Table 5 Summary
13 [%I
films
V/o
Whole polymer Fraction AB, C
ketone)
[PSI
moisture
contents
(V/o)
in Meta-poly
(arylene
[%I
ether
73
Ij [%I
[PSI
1698 f 40 2273 + 77 1651 *I52
ketone)
6+1 5*1 7*1
films
Mol. weight CM,) GPC/LALLS (experimental) ‘) x10-3
Gi
v/o
(A”)
(o/o)
M.W. CM,) calculated b, x10-s
20.8 60.8 19.8
70.8kO.4 130.2 _+0.4 66.2+0.6
0.43 0.47 1.06
22.8 f 0.2 60.3 f 0.3 20.4 f 0.3
‘) M, values determined by gel permeation chromatography/low angle laser light scattering technique tall values have an error of + 10%). b, M, values are calculated by using the equation: Vr = AM:, where A = 1.340X 10-l and B = 0.625. ‘) These two samples were used to calculate the constant A. d, The molecular weight of this sample was calculated using the value of A computed above.
J.J. Singh, A. Eftekhari / Molecular weights of polymers
482
A
Bl
Whole
C
Fraction Fig. 4. Comparison between molecular weights determined by GPCfLALLS
M. The comparison between the molecular weights measured by GPC/LALLS technique and the free volume model is illustrated in fig. 4. Eq. (5) should be applicable for the entire range of orthopositronium lifetimes observed in linear polymers (namely, 0.7-4.8 ns). These lifetimes correspond to molecular weights ranging from 4 X 103-4 X 10” amu. 3.2. Test of the model The model was further tested on two atactic polystyrene samples having well defined physical properties. These samples were compression molded, under 20 tons of pressure, into 13 mm diameter x 5 mm thick discs at 393 K. The discs were allowed to cool down to room temperature before positron lifetime measurements were made in them. Using the positron lifetime data in the two samples, the ratio of their molecular weights was calculated and compared with the reported value [ll]. The results are summarized in table 6. The agreement is reasonably good.
Table 6 Comparison between the experimenatal Sample No.
1 2
M, (*5%)
13~~ 96000
and PLS methods.
4. Discussion Several authors have previously investigated the molecular weight dependence of positron decay rates in polymers. Chuang et al. 1121studied positron annihilation in polypropylene of various weight-averaged molecular weights and found a linear decrease in pickoff annihilation lifetime with increasing molecular weights. West et al. 1131,on the other hand, noted that pick-off annihilation lifetime in polystyrene samples is essentially independent of their molecular weights in the range 103-lo6 amu, in contrast to the situation in Teflon. We have conducted a study of pick-off annihilation lifetimes in amorphous linear polymers with molecular weights in range of 1.5 x 104-1.5 X lo5 amu. Instead of the pick-off annihilation lifetime, we have selected the average free volume cell size (V,) as the parameter of interest, since it is more sensitive to molecular weight variation than the lifetime. From the data described earlier, it appears that V, values correlate with the M, values of the test poly-
and reported values of the molecular weights of atactic polystyrene samples 73
[PSI
1980+ 5 1780 + 10
Ratio of Molecular Weights a)
V,(A3)
9.5.0* 0.5 76.9 + 0.9
a) Calculations of ratio of molecular weights: Vf = AMB and therefore
V
2
= (2)‘: 2
MI/M,
(!&)1/B
1.35-+0.14
1.41t0.04
so we get (F)1/B f2
= M,, where B = 0.625. M2
J.J. Singh, A. Eftekhari / Molecular weights of polymers Table 7 Summary of Mark-Houwink Sample
Para-PAEK Meta-PAEK
and free volume model constants
Mark-Houwink equation a)
Free volume model b,
K
a
A
B
1.20x10-” 9.00x 1o-4
0.60 0.60
2.70~ 1tX2 1.34x10-’
0.625 0.625
‘) Mark-Houwink equation: n = KM”. b, Free volume modeI: V, = AMB.
mers quite well. As expected, a unique set of A and B values predict the test polymer M, values as long as their molecular structure remains unchanged. A change in structure as one goes from Para-PAEK to MetaPAEK geometry, necessitates a different set of constants. A notable feature is the fact that V, values are much larger in the meta geometry than the para geometry. This, however, is quite consistent with the different packing behavior of the two geometries. A further corroboration of the validity of the free volume model was obtained by comparing the MarkHouwink equation [5] and the free volume model constants. The results are summarized in table 7. Clearly, the M-H constants and the free volume model constants exhibit similar behavior when the repeat unit geometries change.
5. Concluding remarks It has been shown that positron lifetime spectroscopy may provide a reasonably accurate technique for measuring molecular weights of amorphous linear polymers. The proposed technique relates the free volume cell size (I’,) with the sample molecular weight (hl) in a manner remarkably similar to that obtained by Mark-Houwink (M-H) between the inherent viscosity (7) and molecular weight (M) of the polymer in solution. However, the M-H technique requires the sample to be in liquid form. The PLS technique, on the other hand, is applicable to samples in solid phase. It has been verified for two geometries of the PAEK where it predicts, with reasonable accuracy, not only the higher l$ values (i.e. looser packing) for the meta
483
geometry, but also the anticipated structural differences between the two geometries, The technique has been further tested on two polystyrene samples where it predicts the ratio of their molecular weights with reasonable accuracy.
Acknowledgements
The authors are grateful to Dr. Jeff A. Hinkley of Langley Research Center for preparing PAEK films and providing information about their physical properties and to Professor Phillips L. Jones of Duke University for supplying polystyrene samples used in this study.
References [II J.J. Singh, A. Eftekhari, B.T. Upchurch and KS. Burns, in: Proc. 3rd Int. Conf. on Positron and Positronium Chemistry, ed. Y.C. Jean (World Scientific, Singapore, 1990) p. 54. 121 Y.C. Jean, Microchem. Journal, 42 (1990) 72. 131Y.C. Jean, in: Proc. 3rd Int. Conf. on Positron and Positronium Chemistry, ed. Y.C. Jean (World Scientific, Singapore, 1990) p. 1. [41 J.J. Singh, A. Eftekhari and T.L. St. Clair, A Study of Physical Properties of ODPA-p-PDA Polyimide Films, NASA Technical Memorandum 102625 (1990). 151 R.B. Seymour and C.E. Carraher, Structure-Property Relationships in Polymers (Plenum, New York, 1984) p. 104. [61 J.A. Hinkley, K.A. Crook and J.R.J. Davis, High Performance Polymers 1 (1988) 61. [71 J.J. Singh, A. Eftekhari and T.L. St. Clair, Nucl. Instr. and Meth. B53 (1991) 342. [81 J.J. Singh, G.H. Mall and D.R. Sprinkle, Analysis of Positron Lifetime Spectra in Polymers, NASA TP-2853 (1988). [91 P. Kirkegaard, Comput. Phys. Commun. 7 (1974) 401. m H. Nakanishi, S.J. Wang, and Y.C. Jean, in: Positron Annihilation in Fluids, ed. S.C. Sharma (World Scientific, Singapore, 1988) p. 292. [111 A.D. Kasbekar, P.L. Jones and A. Crowson, J. Polym. Sci., Polym. Chem. 27 (1989) 1373. ml S.Y. Chuang and S.T. Tao, J. Appl. Phys. 44 (1973) 5171. [I31 D.H.D. West, V.J. McBrierty and C.F.G. Delany, Appl. Phys. (1975) 171.