Free-Volume Hole Distribution of Polymers Probed by Positron

Chapter 27

Free-Volume Hole Distribution of Polymers Probed by Positron Annihilation Spectroscopy J. Liu, Q. Deng, H. Shi, and Y. C. Jean

1

Downloaded by UNIV LAVAL on May 11, 2016 | http://pubs.acs.org Publication Date: May 5, 1995 | doi: 10.1021/bk-1995-0598.ch027

Department of Chemistry, University of Missouri, Kansas City, MO 64110

Positron annihilation spectroscopy (PAS) has been developed to characterize the free-volume properties of polymers and polymer blends. Positron arinihilation lifetime (PAL) measurements give direct information about the dimension, content, and hole-size distributions offreevolume in amorphous polymeric materials. The free-volume hole distribution in epoxy, polypropylene, polycarbonates, and polystyrene is presented as a function of pressure, molecular relaxation, and miscibility. The unique capability of PAS to probefree-volumeproperties derivesfromthe fact that the positronium atom (Ps) is preferentially trapped in atomic-scale holes ranging in sizefrom1 to 10 Å.

The concept of free volume in a liquid, proposed four decades ago [1], has been applied to explain the free-volume hole properties of polymers in recent years [2]. In 1959, Cohen and Turnbull [3] used free-volume hole size to explain diffusion in liquids as a function of temperature and pressure and defined free volume to be the volume difference between the space of the molecular cage made by the surrounding molecules and the van der Waals volume of the molecule in the cage. Free-volume parameters of polymeric materials have recently been calculated using theories such as molecular dynamics and kinetics [4, 5]. Despite the existence of various free-volume theories, only limited experimental data about free volumes in polymers have been reported due to the intrinsic difficulties of probing free volume, which has a size of a few angstroms and a duration as brief as a few 10" s. However, a great deal of effort has been put into measuring freevolume properties in polymeric materials. Small-angle X-ray and neutron diffractions have been used to determine density fluctuations and then to deduce free-volume size distributions [6-8]. A photochromic labeling technique by site10

1

Corresponding author 0097-6156/95/0598-0458$12.00/0 © 1995 American Chemical Society Urban and Provder; Multidimensional Spectroscopy of Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1995.


27. LIU ET AL.

Free-Volume Hole Distribution of Polymers

459

specific probe has also been developed [9-12]. In recent years, a new microanalytical probe, Positron Annihilation Spectroscopy (PAS), has been developed to probe the free-volume properties in polymeric materials. In 1986, we reported [13] the mean free-volume size in an epoxy by measuring its positron annihilation lifetime (PAL) based on a spherical model developed for die freevolume bubbles in liquids [14]. In recent experiments [15-17], using the P A L method we have determined the free-volume hole size in polymers as a function of pressure, temperature, and physical aging. We have also extended our study of free-volume properties to polymer blends [18]. The unique sensitivity of PAS in probing free-volume properties is due to the fact that positronium (a bound atom consisting of an electron and a positron) is found to be preferentially localized in the free-volume region of polymeric materials. Evidence of Ps-localization in free volumes has been found from temperature-, pressure-, and crystallinity-dependent experiments [13, 15, 16]: (1) o-Ps (triplet Ps) lifetime undergoes a dramatic change as Τ > T (glass transition temperature) and Τ < T ; (2) the lifetime temperature coefficient (« 10" K* ) is one order of magnitude larger than the volume expansion coefficients (« ΙΟ K" ); (3) a large variation of positron lifetime has been observed when a polymer is under a static pressure; and (4) o-Ps formation is found only in amorphous regions where free volume exists. In contrast to other techniques, PAS probes the free-volume properties directly without significant interference from bulk properties. g

3

1

g

-4

1

Mean Free-Volume Hole Sizes In P A L measurements, the experimental measured positron annihilation rate λ is defined by the integration of the overlap between the positron density p (r) and the electron density p.(r): +

(1) where the constant is a normalization constant related to the number of electrons involved in the annihilation process. The annihilation lifetime τ is the reciprocal of the annihilation rate λ. In polymeric materials, the positron lifetime spectrum reveals a function containing a multiexponential: N( t) = £

Ie

(2)

±

ί=1,η where η is the number of exponential terms, and \ and λ{ represent the number of positrons (intensity) and the positron annihilation rate, respectively, for the annihilation from the ith state. It is customary to fit P A L spectra into three or four lifetime components by using a computer program (PATFIT) [19]. The longest lifetime component is contributed from o-Ps (ortho-positronium) annihilation. Ps is considered to be confined in spaces between molecules. In order to determine the size of free-volume holes from the o-Ps lifetime, generally we assume the free-volume hole is in a spherical shape under isotropic conditions. By developing equation (1) based on a simple particle-in-spherical-box x

Urban and Provder; Multidimensional Spectroscopy of Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1995.

460

MULTIDIMENSIONAL SPECTROSCOPY OF POLYMERS

quantum mechanical model [20, 21], a correlation between the o-Ps armihilation lifetime (τ) and hole radius (R) has been obtained:


(3)

where R = R „ - a R is the radius of the free-volume hole and a R = 1.656 À is an empirical parameter which was obtained by fitting the measured aririihilation lifetimes of cavities with known sizes [22]. The universal correlation between o-Ps and free-volume is shown in Fig. 1. The mean free-volume hole size of polymers can be easily determined by measuring the o-Ps lifetime and converting the mean lifetime of o-Ps obtained by conventional methods of data analysis into hole size according to the semiempirical equation (3). In the temperature study on a series of epoxy polymers D G E B A / D A B / D D H in different chemical compositions, we found that the o-Ps lifetime increases dramatically at T [13]. One of these lifetime/temperature variations is shown in Fig. 2. It is seen in Fig. 2 that the largest change of o-Ps lifetime coincides with T and the temperature coefficient is on the order of 10" K ' , which is about one order of magnitude larger than the volume expansion coefficient. Furthermore, the temperature dependence also occurs at Τ < T . This effect has been used to probe the sub-T annealing and the physical aging effects on the polymeric structures [23, 24]. However, it is known that the free-volume hole is not exactly spherical, especially under anisotropic conditions. Therefore, we need to consider the shape factor. Under anisotropic external forces, such as anisotropic pressure, stretching, etc., the free-volume hole shape is more likely to be ellipsoidal. We have recently developed a new relationship between the o-Ps lifetime and the anisotropic dimension of an ellipsoidal free-volume hole using an ellipsoidal hole model [25, 26]. In an ellipsoidal hole model, an ellipsoid is defined by the dimensions a and b with respect to the semi-major and the semi-minor elliptic axes and by the eccentricity, ε =(a -b ) /a. Solving the Ps wave function, Φρ , inside the ellipsoid (Φρ = 0 outside the ellipsoid) from the Schrôdinger equation by a numerical method, we obtain the o-Ps lifetime in a manner similar to that used for the spherical model. In practice, we fit the numerical results into a variety of mathematical forms, including polynomials and exponentials. The following equation is obtained for the ratio of o-Ps lifetimes as a function of the eccentricity (β): g

g

3

1

g

g

2

2

1/2

8

8

2

=1+0 . 4 0 0 ε - 4 . 1 6 ε + 2 . 7 6 ε

3

(4)

x

sph

The above equation shows that the o-Ps lifetime in an ellipsoid is always shorter than in a sphere with the same hole volume. A n ellipsoid with a larger



27. LIU ET A L


461

Fig. 1. A universal correlation between o-Ps lifetime and free volumes in polymers. The solid line is the best fitted equation (3) to known cavity volumes [22]. Reprinted by permission of Ref. 22 (© 1988 World Scientific).


462


0.250 Temperature cycle: Down-* Up-*Down


α

δ

•

2.5

- 0.175

c

CO Ε

Φ

.i

0.100 §

2.0

§ φ

CO Q_

ο I 7 = 52 °C g

0.030

1.5

1.0 -100

0

100 Temperature (°C)

200

Fig. 2. Temperature dépendance of o-Ps lifetime and free volume in an amine-cure epoxy polymer (T = 52 °C) [13]. Four onset temperatures were observed: T (solidified), T ' (real glass transition), T (apparent glass transition), T (Ps-bubble formation). Reprinted by permission of Ref. 13 (© 1986 J. Polym. Sel B). g

s

g

g

e


27. LIU ET AL.

463


eccentricity gives a larger reduction of o-Ps lifetime. Eq. (4) can be used to correct the shape factor from P A L results. Free-Volume Hole Distributions In reality there is a range of free-volume hole sizes in polymeric materials. Therefore, it is more adequate to express o-Ps lifetime in a distribution form rather than as a discrete value, τ . Hence, we employ the expression of positron lifetime spectra in the form originally suggested by Tao and later by Schrader [27] as: 3

kt

N(t) =f~ka(X)e- dk+B

(5)


Jo

where λα(λ) is an annihilation probability density function and Β is the background of the spectrum. The exact solution of λα(λ) is a very difficult problem because the actual experimental spectra are convoluted with a resolution function which cannot be exactly measured. However, i f one measures a reference spectrum Y (t), which has a known single positron decay rate (λ ), the solution of λα(λ) can be obtained by using Y (t) to deconvolute sample spectra. Here, Y (t) can be expressed as: r

Γ

r

Y (t) z

r

klt

=R(t) *N X e' r

(6)

r

where N is the normalized total count for a reference spectrum and R(t) is the resolution function. Lifetime spectra of reference samples were obtained from the following materials: a well-annealed single ΑΙ (τ = 162 ps, 98%) and a B i radioisotope which emits two γ-rays at τ = 183 ps with energies very close to ^ a . The computer program CONTIN [28, 29] was used to perform the deconvolution procedure through a Laplace inversion technique. In this continuous data analysis method, a result of λα(λ) vs λ is obtained from spectra. A l l reference spectra give consistent λα(λ) vs λ results. Following the correlation Eq. (3) between τ and hole radius R, and considering the difference of o-Ps capture probability in different hole size with a linear correction Κ (R) (=1+8R) and with a spherical approximation of free-volume holes, the free-volume hole volume probability density function, Vfpdf is expressed as [30]: r

2 0 7

3

V pdf =-3. 32 f

[

C

O

B

(

,

1

]

a

(

A

^ 4 6 " fi+1.66 (R+1.66) K(R)4nR 2

)

{

(7) 7

)

2

The free-volume distributions in polymers obtained by P A L measurements using continuous data analysis are compared quantitatively with the SimhaSomcynsky theory [31]. A good agreement between our experimental results and predictions of the theory is a further indication that PAS is a reliable and sensitive probe to determine the free-volume distributions in polymeric materials.


464


Pressure Dependence of Free-Volume Hole Distributions The investigation of pressure/volume behavior of polymers is essential to the fundamental understanding of material properties. In the microscopic point of view, the changes in macro-physical properties due to applied pressure are related to the changes in free-volume properties. In the study of pressure dependence of free-volume properties in polymers, we have used a thermosetting polymer, epoxy, and a thermoplastic polymer, polypropylene [30, 32]. The positron lifetime distributions of epoxy and polypropylene are plotted in Fig. 3 under different pressures. Each spectrum contains a total statistics of 40 χ 10 counts. The good agreements between the mean lifetime results from a least-squares fit method (PATFIT) and the peak lifetimes from a Laplace inverting method (CONTIN) as shown in Fig. 3 confirm the accuracy of our new method of data analysis. The new information here is a continuous lifetime distribution instead of discrete lifetimes. The free-volume hole radius distributions are then obtained by converting the o-Ps lifetimes (right peaks) into R according to the τ-R correlation of Eq. (3), then converting to volume, V , according to Eq. (7). The results of hole volume distributions are shown in Fig. 4. As expected, an increase of pressure results in collapse and compression of the free-volume holes. In epoxy the hole distributions shift from maxima 50 Â to 25 A , 8 À , and 1 À for 1.8, 4.9, and 14.0 kbar respectively, while in polypropylene the maxima shift from 100 A to 25 A and 7 À for 4.2 and 14.7 kbar. The width of the distribution is also correspondingly narrowed as Ρ increases. We found that the experimental hole distributions can be fitted into gaussian function or with a sum of gaussians. The compressibility of free volume, calculated from the peak values of free-volume distributions under different pressures, is in the range of 10' to 10" at low pressures. It is found to be a few times larger than the results obtained from a volumetric method [2]. We believe that our p value is more accurate than those obtained by the conventional methods because PAS is probing the free volume directly. It is worthwhile to mention that a distorted distribution of very small holes, clearly seen in Fig. 3 for Ρ of ca. 14 kbar, is a result of using a crude theoretical quantum mechanical model for Pslocalization in the free-volume holes. It also shows a limiting radius of holes, ~ 0.5 Â, that Ps can probe properly.


6

3

3

3

3

3

5

3

3

6

f

Free-Volume Hole Distributions and Molecular Relaxation A non-equilibrium state often exists in glassy polymers resulting from the thermal or mechanical treatment history. Glassy polymers exhibit a tendency toward equilibrium state in order to minimize the overall energy through molecular relaxation of polymer chains over a period of time. The chain mobility of macromolecules in a closely packed system is primarily determined by and inversely proportional to the degree of packing of the system. At the molecular level, the chain mobility is restricted by the free-volume hole size and hole size distribution. On the other hand, the free-volume properties are changed through molecular relaxation. Hence, a very fruitful approach to understanding the


27. LIU ET AL.

465

Free-Volume Hole Distribution of Polymers PRESSURE DEPENDENCE OF EPOXY POLYMER

6.6 · Ρ -

0.0 Debar)

Ρ -

L8 (kbar)

Ρ -

4.0 (kbar)

3.3

ft/ \ .

0.0 6.6 3.3 -

ο W

0.0 6.6

01

1


3.3 0.0 6.6

* ·· • · · ··-·-• ··· · ··-·

·.····..

,

.1

,

>

f

t

ι

Ρ - U.0 (kbar) 3.3

H• · ·

0.0

.

·*·······

*-* 0.0

1——ι 0.5

•

1

«

1

«

1.0 1.5 2.0 UFETIUE r (nsec)

1 2.5

3.0

PRESSURE DEPEKOENCE 0Γ POLYPROPYLENE

A-

Ρ -

0.0 (kbar)

3 21 0 5

4-

4.2 (kbar)

3 2 1 0 5 4-

Ρ - 14.7 (kbar)

3 2 1 0

Q.O

—I 0.5

1.0

I

1.5 LIFETIME

—I 2.0 τ (nsec)

2.5

—r— 3.0

3.5

Fig. 3. Positron lifetime distribution functions of (a) epoxy, (b) polypropylene. The applied pressure is quasi-isotropic [30, 32]. The distributions of positron lifetime were obtained by using the Laplace inverting program CONTIN. Reprinted by permission of Refs. 30 (® 1992 J. Polym Sci. B) and 32 (© 1992 ACS).



466


PRESSURE DEPENDENCE OF POLYPROPYLENE 10

I ' ι » ι I ι ι ι ι I ι ι ι ι I ι t r ι I ι ι ι ι I ι—l-l

FREE-VOLUME

(A°)

Fig. 4. Free-volume hole volume distribution functions of (a) epoxy, (b) polypropylene, under different pressures. Smooth lines were drawn through data for eye-guide purposes only [30, 32]. Reprinted by permission of Refs. 30 (© 1992 J. Polym Sel B) and 32 (© 1992 ACS).


27.

LIU ET AL.

molecular relaxation mechanism is to investigate the microscopic free-volume properties as a function of time after polymeric materials are thermally or mechanically treated [1]. Recently, we have studied the epoxy and polypropylene polymers for freevolume distribution as a function of time after polymers are released from a static pressure [33]. The free-volume hole distributions of epoxy and polypropylene are shown in Fig. 5 before and after press. As shown in Fig. 5, the free-volume hole volume distributions are found to be observably different for polymers before and after press. The distributions become narrower after applying 14.7 kbar of pressure for one week. The distribution in epoxy can be approximately expressed by gaussian type functions with F W H M (Full Width at Half Maximum) of 60 A and 50 A before and after press respectively, i.e. 17 % narrower after press. In polypropylene, F W H M are found to be 105 Â and 65 À before and after press respectively, i.e. 38 % narrower after press. These results show that applying an external pressure has significantly changed the microstructure of polymers. Increasing pressure causes a compression or collapse of free-volume holes. Upon the release of pressure, a major fraction of the molecular chains, where the free volume is at a maximum distribution (le. the peak of the distribution), relaxes spontaneously back to the original size. Part of the molecular chain relaxes at a slower rate and is detectable from the current PAS method. As shown in Fig. 5, in epoxy, after 75 days the free-volume distribution recovers gradually back to the original distribution. In polypropylene, we found the difference of hole distribution before and after press is much greater than that in epoxy. On the other hand, we found the result of the distribution is not changed as a function of time up to 75 days of aging, i.e. the distributions are all the same as that at 10 days, as shown in Fig. 5. This indicates that a fraction of molecular chains in polypropylene relaxes at a rate much slower than in epoxy. This slower molecular relaxation in polypropylene than in epoxy after press may be due to the difference in the polymer structures: polypropylene is a semicrystalline sample and epoxy is 100% amorphous; polypropylene is a thermoplastic polymer and epoxy is a thermosetting polymer. These differences in thermal and mechanical properties between these polymers result in a different response in the molecular relaxation after the release of pressure. It appears that deformation of free-volume holes can be easily recovered in thermosetting polymers but not in thermoplastic materials. 3


467


3

3

3

Free-Volume Hole Distributions and Miscibility In comparison with their neat polymer components, the morphology of polymer blends is more complicated due to the process of mixing. A variety of interesting new features in polymer blends can be generated according to the equilibrium thermodynamics and kinetics of mixing. Based on thermodynamics, a negative enthalpy of mixing, Δ Η ^ , is often required for polymer blends to contribute to lower the free energy of mixing, A G ^ , due to the small and negligible entropy of mixing, A S ^ [34]. As a result, the majority of polymer blends are immiscible. The phase separation of immiscible polymer blends creates different domains as well as interfacial regions. The heterogeneity of the morphology resulting from the phase separation has a significant effect on physical properties of blends. In our Urban and Provder; Multidimensional Spectroscopy of Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1995.



468

Fig. 5. Free-volume hole volume distributions at 25 °C before and after press (10 and 75 days after releasing pressure) (a) epoxy, (b) polypropylene [33]. Reprinted by permission of Ref. 33 (® 1992 ACS).


27. LIU ET AL.

469



study of the change of free-volume distributions corresponding to the miscibility of polymer blends, we investigated two different types of polymer blends: a miscible blend of bisphenol-A polycarbonate (PC) and tetramethyl bisphenol-A polycarbonate (TMPC), and an immiscible blend of PC and polystyrene (PS) [18]. As shown in Fig. 6, the differences in free-volume distributions can be seen for the two types of blends. A larger difference between the blends and the pure polymers in the distribution of free-volume holes is observed in the immiscible blend (PS/PC) than in the miscible blend (TMPC/PC). In the immiscible blend, the distribution is broader than in the pure polymers due to the free volumes formed in the interfacial regions. Further systematic investigation on the free-volume effect using the P A L method on blends is in progress at our laboratory. 25 20

Ό -α

TMPC-PC

15 10

D

5 0

50

100

150

200

250

300

3

V (A ) r

25 ^

20

^ Ο

15

ι—I

% a.

10

>

5 0

150

200

250

300

3

V (A ) f

Fig. 6. Free-volume hole volume distributions in the TMPC/PC (miscible) and PS/PC (immiscible) polymer blends [18]. Reprinted by permission of Ref. 18 (© 1994 Materials Research Society).


470



Conclusion We have presented the PAS technique as a new microanalytical probe for the characterization of free-volume properties in polymers and polymer blends. The unique behavior of Ps localization in free-volume holes of polymers enables us to use PAS to determine the hole size, hole size distribution function, and concentrations of free volumes at atomic scales in polymeric materials. P A S is a novel probe with applications not only to polymers but also to other technologically important materials, such as pores in catalytic materials and surface states of solids. Four major developments in PAS will be very beneficial in the future: (1) improvement of P A L data analysis into a lifetime vs. amplitude spectrum as in conventional spectroscopy, so that a hole size distribution function can be obtained more accurately; (2) development of two-dimensional A C A R spectroscopy for polymeric applications, so that a detailed three-dimensional hole structure may be mapped out; (3) development of a monoenergetic slow positron beam, so that a direct application to thin film polymers can be made; and (4) development of a universal correlation equation including chemical quenchings. Acknowledgement This research was supported by a grant from the National Science Foundation (DMR-90040803). Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Doolittle, A. K. J. Appl. Phys., 1951, 22, 1471. For example, see Ferry, J. D. Viscoelastic Properties of Polymers, 3rd. ed. Wiley, N.Y. 1980. Cohen, M. H.; and Trunbull, D. J. Chem. Phys. 1959, 31, 1164. Takenchi, H.; and Roe, R. J. J. Chem. Phys. 1991, 94, 7446. Robertson, R. E.; Simha, R.; and Curro, J. G. Macromolecules 1988, 18, 2239. Flouda, G.; Pakula, T.; Stamm, M.; and Fisher, E . W. Macromolecules 1993, 26, 1671. Song, H. H.; and Roe, R. J. Macromolecules 1987, 20, 2723. Nojima, S.; Roe, R. J.; Rigby, D.; and Han, C. C. Macromolecues 1990, 23, 4305. Hooker, J. C.; Royal, J. S.; and Torkelson, J. M. Polym. Prepr. 1993, 34, 498. Winudel, M. B.; and Torkelson, J. M. Polym. Prepr. 1993, 34, 500. Royal, J. S.; Victor, J. G.; and Torkelson, J. M. Macromolelues 1992, 25, 729. Royal, J. S.; Victor, J. G.; and Torkelson, J. M. Macromolelues 1992, 25, 4792. Jean, Y. C.; Sandreczki, T. C.; and Ames, D. P. J. Poly. Sci. B. 1986, 24, 1247. Ferrell, R. A. Phys. Rev. 1957, 108, 167.


27. LIU ET AL. 15. 16. 17.

18.


19. 20. 21. 22.

23. 24. 25.

26. 27. 28. 29. 30. 31. 32. 33.

34.


471

Nakanishi, H.; Jean, Y. C.; Smith, E. G.; and Sandreczki, T. C. J. Poly. Sci. B. 1989, 27, 1419. Wang, Y. Y.; Nakanishi, H.; Jean, Y. C.; and Sandreczki, T. C. J. Poly. Sci. B. 1990, 28, 1431. Jean, Y. C.; Zandiehnadem, F.; and Deng, Q. in Proc. of MRS Symp. Structure, Relaxation, and Physical Aging of Glassy Polymers, vol. 215, pp. 163-174 (R.J. Roe and J.M. O'Reilly, Eds.) MRS Pub., Pittsburgh, PA (1991). Liu, J.; Jean, Y. C.; and Yang, H. in Proc. of MRS Symp. Crystallization and Related Phenomena in Amorphous Materials, vol. 321 (Libera, M.; Cebe, P.; Dickinson Jr., J. E.; Eds.) MRS Pub., Pittsburgh, PA (1994), p. 47. A PATFIT package was purchased from R i s ØNational laboratory, Roskilde, Denmark. Tao, S. J. J. Chem Phys. 1972, 56, 5499. Eldrup, M.; Lightbody, D.; and Sherwood, J. N. Chem Phys. 1981, 63, 51. Nakanishi, H.; and Jean, Y. C. in Positron and Positronium Chemistry, (D.M. Schrader and Y. C. Jean, Eds.) Elsevier Pub. Amsterdam (1988), Chapter 5. Sandreczki, T. C.; Nakanishi, H.; and Jean, Y. C. in Proc. of Int. Symp. in Positron Annihilation Studies of Fluids, (S. C. Sharma, Ed.) World Sci. Pub., Singapore (1988) p. 200. Kobayashi, Y.; Zeng, W.; Meyer, E. F.; McGervey, J. D.; Jamieson, A. M.; and Simha, R. Macromolecules, 1989, 22, 2302. Jean, Y. C.; Shi, H.; Dai, G. H.; Huang, C. M.; and Liu, J. in Proc. of 10th Conf. on Positron Annihilation, (He, Y. J.; Cao, B. S.; Jean, Y. C. Eds) Trans Tech pub., Beijing, China (1994) p. 691. Jean, Y. C.; and Shi, H. J. Non-Crys. Solid., in press (1994). Tao, S. J. IEEE Trans. Nucl. Sci., 1968, 15, 175; Schrader, D. M. in Positron Annihilation, pp. 912-914 (P. G. Coleman, S. C. Sharma and L. M. Diana, Eds.) North-Holland, Amsterdam (1982). Provencher, S. W. CONTIN Users Manual, EMBL Technical Report DA05, European Molecular Biology Laboratory, Heidelberg, Germany Gregory, R.B.; and Zhu, Y. Nucl. Instr.Meth.Phys., 1990, A290, 172. Jean, Y. C.; and Deng, Q. J. Polym. Sci. B, Polym. Phys. 1992, 29, 1359. Liu, J.; Deng, Q.; and Jean, Y. C. Macromolecules 1993, 26, 7149. Deng, Q.; and Jean, Y. C. Macromolecules 1993, 26, 30. For example, see Deng, Q., in Ph.D. Dissertation: Characterization of Free-Volume Properties in Polymers by Positron Annihilation Spectroscopy, University of Missouri-Kansas City, Kansas City, Missouri, 1993. Paul, D. R.; and Sperling, L. H.; Eds. Multicomponent Polymeric Materials; American Chemical Society: Washington, D. C. 1986.

RECEIVED February

2, 1995


Free-Volume Hole Distribution of Polymers Probed by Positron

Recommend Documents