Macromol. Chem. Phys. 199,1917-1926 (1998) 1917

Crystallization analysis fractionation (CRY STAF) of poly(ethy1ene-co-1-octene) made with single-site-type catalysts: A mathematical model for the dependence of composition distribution on molecular weight

Jo60 B. I? Soares*’, Benjamin Monrabap, Jesu‘s Nieto3, Javier Blanco4

Institute for Polymer Research, Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

* Polymer Char, Valbncia Parc Tecnolbgic, P.O. Box 176, E-46980 Paterna, Spain Dow Chemical, Polyolefins Research, B-1470,2301 N. Brazosport Blvd., Freeport, Texas 77541, USA Dow Chemical, Analytical Laboratory, P.O. Box 195,43080 Tarragona, Spain

(Received: December 18, 1997; revised: March 23, 1998)

SUMMARY A series of seven poly(ethy1ene-co- 1 -octene) samples made with a single-site-type catalyst were analyzed by crystallization analysis fractionation (CRYSTAF). The samples have similar average copo- lymer compositions but differ in their average molecular weights. CRYSTAF results confirm the theoretical predictions of Stockmayer’s distribution: the breadth of the chemical composition distribution is inversely proportional to the number average molecular weight for polymers made with single-site-type catalysts. Stockmayer’s distribution was used to assess the instrumental peak broadening of CRYSTAF. An extension of this treatment for polyolefins made with multiple-site-type catalysts is proposed.

Introduction Stockmayer’s bivariate distribution has been used to model the ideal fractionation of polyolefins by tempera- ture rising elution fractionation (TREF)’). The term ideal fractionation is used to indicate that the separation mechanism is assumed to be controlled only by copoly- mer composition or stereoregularity. Cocrystallization effects, molecular weight influences and peak broadening due to axial dispersion and crystallization kinetics are considered negligible. These hypotheses clearly limit the application of the model, since peak broadening in TREF can be quite ~ignificant~.~).

Until recently, it was very difficult to assess the instru- mental peak broadening in TREF, since polyolefins made with heterogeneous Ziegler-Natta catalysts have broad and often multimodal chemical composition distributions caused by the presence of multiple site types on the cata- lyst surface. The complexity of the TREF curves makes it difficult to isolate instrumental peak broadening effects from broadening caused by the presence of different site types on the catalyst. The same observations are valid for crystallization analysis fractionation (CRYSTAF). CRYS- TAF is a novel analytical and preparative technique that generates results equivalent to the ones obtained with TREF at a much shorter analysis time and operational costs49 10).

However, single-site-type catalysts such as metallocene and constrained geometry catalysts can produce polyole- fins with narrow chemical composition and molecular weight distribution^^.^). It has been shown that the mole-

cular weight distribution of these polyolefins follow Flory’s most probable dis t r ib~t ion~*~-~) . It is reasonable to assume that their chemical composition distribution will also obey Stockmayer’s bivariate distribution. (Stock- mayer’s bivariate distribution reduces to Flory’s ilistribu- tion if it is integrated over all copolymer compositions.) These novel polyolefins are ideally suited to investigate TREF and CRYSTAF fractionation, since the instrumen- tal responses of these techniques can be compared with Stockmayer’s distribution to evaluate instrumental peak broadening.

In this article, Stockmayer’s distribution is used to interpret the CRYSTAF analytical results of seven poly- (ethylene-co- 1-octene) samples made with a single-site- type catalyst. It is expected that the proposed methodol- ogy is equally applicable to TREE

Theoretical background Stockmayer’s bivariate distribution is defined for linear binary copolymers by the following equations”):

1 w(r,y)drdy = r z2 exp(-r z)dr

where r is polymer chain length, y is the deviation from the average mole fraction, Fl, of monomer type 1 (ethy-

Macromol. Chem. Phys. 199, No. 9 0 WILEY-VCH Verlag GmbH, D-69451 Weinheim 1998 1022- 1352/98/0909-1917$17.50+.50/0

1918 J. B. P. Soares, B. Monrabal, J. Nieto, J. Blanco

So00

7000

6000

5000

g 4-

3000

2000

loo0

0 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

Deviation fiom average copolymer composition (y)

Fig. 1. CCD component of Stockmayer’s bivariate distribution for several values of the parameter j?~

lene) in the copolymer, and rl and r2 are reactivity ratios for copolymerization.

The ratio of transfer to propagation rates, z, can be cal- culated using the following generic equation:

Notice that, for single-site-type catalysts, the number and weight average chain lengths, r, and r,, are simply given by:

(4)

which results in a polydispersity index (PDI, defined as rwlr,) of 2.

Eq. (1) can be used to calculate the chemical composi- tion distribution (CCD) for copolymer of a given chain length r. It is evident that the CCD component of Q. (1) is Gaussian with a variance 0’ equal to /3/. Consequently, the CCD for a given chain length and average composi- tion will become broader with increasing rl r2, i. e., with increasing tendency to form block copolymers. Similarly, CCDs of shorter chains will be necessarily broader than CCDs of longer chains.

Eq. (1) can be integrated for all chain lengths to obtain the CCD component of Stockmayer’s distribution:

r- 2

Eq. (5) can be used to predict the CCD of binary olefin copolymers made on single-site-type catalysts. An exten- sion of this model for polyolefins made with multiple-site- type catalysts has been published by Soares and Hamielec’).

According to Eq. (3, the CCD becomes broader as the product of the parameters p and z increases, i. e., for poly- mers with smaller r, and higher tendency to form long sequences of one monomer type. These trends are illu- strated in Fig. 1.

Both TREF and CRYSTAF are presumably not affected by molecular weight, besides the broadening effect described above. Considering CRYSTAF (or TREF) frac- tionation of single-site-type polyolefins to be governed solely by chain crystallinity and assuming that chain crys- tallinity can be directly translated into copolymer compo- sition using a calibration curve, the CCD measured by both of these techniques should follow Eq. (5). However, instrumental peak broadening might occur due to fractio- nation non-idealities, such as cocrystallization and crys- tallization kinetics effects. This would lead to measured distributions that are broader than those predicted by Stockmayer ’s distribution.

Crystallization analysis fractionation (CRYSTAF) of poly(ethy1ene-co- 1 -octene) made with single-site-type catalysts ... 1919

Most rigorous methods for correcting instrumental peak broadening make use of the following integral equa- tion12. 16).

where w(y) is the actual distribution (in our case given by Eq. (5 ) ) , G ( 9 , y ) is the instrumental spreading function, and F ( j ) is the measured distribution. Several different instrumental spreading functions have been proposed"). For symmetrical broadening, a Gaussian distribution can be applied:

(7)

For deviations from the Gaussian shape, the following general spreading function has been proposed13. 14):

m

(9)

where g ( 9 , y ) denotes the n"-order derivative of the Gaussian distribution g(3, y) with respect to j .

The coefficient A3 provides a measure of the skewness: distributions skewed to the right have positive A3, distribu- tions skewed to the left have negative A3, and symmetrical distributions have A3 equal to zero. The coefficient A4 pro- vides a measure of the flattening of the distribution: when A4 is positive, the detector response is higher and narrower than a Gaussian distribution, when A4 is negative, the response is flatter and broader than a Gaussian distribu- tion. Symmetrical distributions have A4 equal to zero.

In Eqs. (7) to (9), the parameters of the spreading func- tion can be either constant (uniform spreading) or vary as a function of the variable y (non-uniform spreading).

Eq. (6) will be used in combination with Eqs. (5) , (8) and (9) to investigate instrumental peak broadening in CRY STAF.

Experimental part The seven poly(ethy1ene-co- 1 -octene) samples studied in this work were synthesized with Dow's constrained geome- try catalyst (CGC) using a solution polymerization process in a continuous stirred-tank reactor. Tab. 1 shows the average properties of these resins. The samples can be divided in two groups: samples A and C have a 1-octene mole fraction of about 0.19 while samples D to H have a 1-octene mole frac- tion of about 0.15. The main difference between the samples is their molecular weight averages. Polydispersity indexes are close to the theoretical value of 2.

Tab. 1. Average properties of poly(ethy1ene-co- 1 -octene) samples

Sample Melt Density Content of Average mole M, index in g/cm3 1-octene fraction of

in mol-% ethylene, F ,

A 3 0.902 19.3 0.807 29600 2.20 C 30 0.902 19.7 0.803 17300 2.23

D 0.5 0.910 13.8 0.862 38000 2.21 E 1 0.910 14.3 0.857 34400 2.31 F 3.5 0.910 15.1 0.849 29300 2.15 G 6 0.911 14.8 0.852 22000 2.54 H 30 0.913 15.2 0.848 19800 2.06

I

- 0.9 5 I

0.8 8

f

* E

0.7

0.6 - f

Fig. 2. octene)

CRYSTAF calibration curve for poly(ethylene-co-1-

Molecular weight averages and polydispersities were mea- sured by high-temperature size exclusion chromatography (SEC) using 3 columns of 10 micron mixed bed (500 to 10 million molecular weight range) in a Waters 15OC apparatus at 140°C. Calibration was based on narrow polystyrene stan- dards and applying a polyethylene shift factor.

Average 1-octene contents in the copolymer samples were measured by infrared spectroscopy using a Nicolet 730 FTIR calibrated by I3C NMR, which approximates mass balance results calculated from the polymerization process.

Melt index was measured according to ASTM-D- 1238-95 at 190°C using a 2.16 kg weight. Density was measured according to ASTM-D-792-9 1.

In the CRYSTAF apparatus, manufactured by Poly- merChar S.A., Valencia (Spain), a Hewlett Packarcl 5890 I1 gas chromatograph is used to perform the crystallization temperature program. The crystallization is carried out in stirred stainless steel containers of 60 ml volume where dis- solution and filtration takes place automatically4). Up to five crystallization containers can be installed in the main oven. The infrared detector used is a dual wave length optoeletro- nic unit with a heated flowthrough cell operating at 150°C and using 3.5 pm as the measuring wavelength. Concentra- tions of 0.1 g/dl were used in this investigation, with 30 mg of polymer samples in 30 ml of 1,2,4-trichlorobenzene sol- vent (commercial grade, redistilled). The crystallization rate was 0.2 "Chin between 90°C and 30 "C for most of the sam- ples. The whole process of dissolution, crystallization and

1920

cleaning was done, as usual, in automatic mode. The whole analysis time for five samples takes only six h.

CRYSTAF was calibrated using poly(ethy1ene-co- 1 - octene) standards with narrow crystallization temperatures as presented in a previous publication"). Fig. 2 shows the cali- bration curve used in this investigation.

Results and discussion Fig. 3 shows CRYSTAF crystallization temperature dis- tributions for samples A to H. The crystallization tem- perature distributions become broader as the number average molecular weight (@,) decreases, in excellent qualitative agreement with the predictions of Stock- mayer 's distribution. Average parameters for these distri- butions are shown in Tab. 2. The dispersion parameters R and CT increase with decreasing M,,, as expected. The cali- bration curve presented in Fig. 2 can be used to obtain the CRYSTAF-measured CCDs shown in Fig. 4.

Stockmayer's CCDs for the samples are shown in Fig. 5. It was assumed that rlr2 = 1 for all samples, since data on reactivity ratios were not available. As seen before, the predicted distributions would be broader for rlr2 > 1 and narrower for rlr2 < 1. Average I-octene frac- tions (l-B,) shown in Tab. 1 were used in the calcula- tions. The parameter z was calculated from the M, values shown in Tab. 1 using the equation:

where mwethylene and mwoctene are the molecular weights of ethylene and 1 -octene, respectively.

J. B. P. Soares, B. Monrabal, J. Nieto, J. Blanco

Tab. 2. CRYSTAF average crystallization temperatures for samples A to H

Sample Twin "C R a i n "C M" A 47.1 C 44.3

D 57.7 E 56.6 F 54.3 G 56.8 H 52.8

1.8 5.6 29 600 5.7 8.7 17300

1 5.1 38000 1.3 5.6 34400 1.4 5.7 29 300 2.4 7.2 22 000 4.8 9.2 19800

Comparing Fig. 4 and 5, one notices that while the gen- eral dependency of CCD on molecular weight is similar for CRYSTAF measurements and Stockmayer 's predic- tions, CRYSTAF distributions are clearly skewed to lower ethylene mole fractions. Additionally, CRYSTAF curves are also broader than the theoretical prediction by Stockmayer's distribution. This is best seen in Fig. 6 for samples A, D and H. Similar trends were obtained for the other polymer samples.

There are two alternative explanations for this observa- tion. First, the CCD of the polyolefin samples might not follow Stockmayer's distribution. This could arise if the polymers were not made with a single-site-type catalyst or if the polymerization took place under non-uniform conditions such as compositional drift and temperature variations. This explanation seems unlikely, since the

16

D

0 10 20 30 40 50 60 70 80

Temp&= 0 Fig. 3. CRYSTAF crystallization temperature profiles for samples A to H

Crystallization analysis fractionation (CRYSTAF) of poly(ethy1ene-co- l-octene) made with single-site-type catalysts ...

30

25

20

1s

10

S

0 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Molar fraction of ethylene (F,)

Fig. 4. CRYSTAF CCDs for samples A to H

1921

45

40

35

30

2s

2 Y 20

1s

10

S

0 0.75 0.77 0.79 0.81 0.83 0.85 0.87 0.89 0.91 0.93 0.95

Molar fraction of ethylene (F, )

Fig. 5. Stockmayer’s CCDs for samples A to H

molecular weight distributions of these polymers Eq. (6) was used to evaluate CRYSTAF peak broaden- approach Flory’s distribution and reactor conditions dur- ing, assuming that w(y) was given by Stockmayer’s CCD ing polymerization were well-controlled and uniform. as given in Eq. (5). Eqs. ( 8 ) and (9) were used as the gen- The second and more likely explanation is that instru- eral spreading function. The parameters of the spreading mental peak broadening is responsible for the observed function were estimated using a Levenberg-Marquardt difference between CRYSTAF profiles and Stockmayer’s method15). It was found that the best form for the instru- predictions at the crystallization conditions used. mental spreading function corresponds to Eiq. (8) with

1922 J. B. P. Soares, B. Monrabal, J. Nieto, J. Blanco

50

h

5 E

SampleA SampleD

A SampleH

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 I Molar &action of ethylene (F,)

Fig. 6. samples A, D and H

Comparison of CCD measured by CRYSTAF and predicted by Stockmayer’s distribution for

20 c I

SampleA

o SampleC

0.6 0.65 0.7 0.75 0.8 0.85 0.9

Molar 6action ofethylene (F , )

Fig. 7. Instrumental spreading function for samples A and C

A3 < 0 (i.e., skewing to lower ethylene mole fractions in the copolymer or lower crystallization temperatures). Inclusion of higher terms, i.e., A4, As, etc., did not improve the fit. The average ethylene fraction in copoly- mer, F , , was also left as an adjustable parameter to account for possible deviations from CRYSTAF calibra- tion curve and the analytical technique used for measur- ing F, for the whole samples. It will be shown that the

fitted F , s agree very well with the measured F , s for the whole polymer samples.

Fig. 7 and Fig. 8 compare the fitted distributions with the original Stockmayer’s and CRYSTAF-measured dis- tributions for samples A and c, and D and H, respec- tively. Similar results were obtained for the other sam- ples. The agreement is reasonably good, although the spreading function does not adequately fit the distribution

Crystallization analysis fractionation (CRYSTAF) of poly(ethy1ene-co- 1 -octene) made with single-site-type catalysts ... 1923

0 SampleD

o SampleH

0.65 0.7 0.75 0.8 0.85 0.9 0.95

Molar ftaetion of ethylene (F,)

Fig. 8. Instrumental spreading function for samples D and H

Tab. 3. (8) and (9)) for samples A to H

Parameters for the uniform spreading functions (Fiqs.

Sample a2 A3 F , ~~~

A 0.000358 -0.00433 0.80 C 0.001498 -0.01012 0.78

D 0.000187 -0.00283 0.86 E 0.000376 -0.00460 0.85 F 0.000385 -0.00462 0.84 G 0.000425 -0.007 13 0.85 H 0.000888 -0.00474 0.84

tail for lower ethylene mole fractions of some samples, notably the ones with higher an. Tab. 3 shows the para- meters for the spreading functions of samples A to H. '

In Fig. 7 and 8, the spreading function was assumed to be uniform, i. e., a* and A3 are assumed to be constant and independent of copolymer composition (or crystallization temperature). It is possible to improve the fit slightly by using non-uniform spreading functions such as

o2 = G$ + @( 1 - F , ) (11)

since the CRYSTAF distributions seem to become broader for lower ethylene mole fractions in the copoly- mer. In Eiq. (1 l), 0 is a positive adjustable parameter and FI is the molar fraction of ethylene in the copolymer. Fig. 9 compares the fit of the uniform and non-uniform spreading functions for sample G. The non-uniform spreading function provides a slightly better fit for lower ethylene mole fractions of the distribution and avoids the bimodality predicted by the uniform spreading function. Several other non-uniform spreading functions were

tested. They are listed in the Appendix. None provided a better fit than the one defined by Q. (1 1). Since the fit is not greatly improved with non-uniform Spreading func- tions and they require at least one more adjustable para- meter, @, the use of non-uniform spreading fiinctions does not seem to be justified for these samples.

From the analysis of Tab. 3, it can be seen that the parameters for the spreading function vary significantly among the several investigated polymer samples. This is also illustrated in Fig. 10. The symmetrical broadening parameter 2 decreases with increasing chain length. On the other hand, the skewing parameter A3 increases with increasing chain length. However, for the samples; with r,, between 500 and 900, these broadening parameters assume values that are approximately the same. Ideally, one would benefit from having the broadening parameters independent of r, to make it possible to correct CRYS- TAF profiles for any polymer sample. An in-deptlh inves- tigation of different spreading functions might overcome this limitation.

The extension of this model for polyolefins made on multiple-site-type catalysts, such as heterogeneous Zieg- ler-Natta catalyst, is straightforward. In this case, polymer CCD can be represented as a weighted average: of the CCDs of polymer chains made on each site type:

i= I

were mi is the weight fraction of polymer made on each site type i, wiCy) is given by Fq. (3, and s is the number of site types on the catalyst.

1924 J. B. P. Soares, B. Monrabal, J. Nieto, J. Blanco

h

5 L

20

18

16

14

12

10

8

6

4

2

0

Fig. 9.

0.0016

0.0014

0.0012

0.001

0.0008 ’b

O.OOO6

0. oO04

0.0002

0

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

Molar hction of ethylene (F,)

Comparison of non-uniform and uniform spreading functions for sample G

0

0

0 0 0

o CRYSTAF

-Uniform

~-N~n-uniform

0

-0.002

-0.004

-0.006

-0.008

-0.01

-0.012

O a t

A,

s

Fig. 10. Parameters a’ and A3 as a function of number average chain length, r,

CRYSTAF profiles for polyolefins made with multiple- A simulated CRYSTAF profile of an olefin copolymer made with a hypothetical five-site-type catalyst is illu- strated in Fig. 11. Stockmayer’s distribution for one site type is also illustrated for comparison. The general spreading function given by Eqs. (8) and (9) with “aver-

site-type catalyts can therefore be described by:

( 1 3 )

Crystallization analysis fractionation (CRYSTAF) of poly(ethy1ene-co- I-octene) made with single-site-type catalysts ... 1925

1 4 1 12

6.00E-01 6.50E-01 7.00E-01 7.5OE-01 8.00E-01 8.5OE-01 9.00E-01 9.5OE-01 1.00E+00

Molar fraction ofethylene (FI)

Fig. 11. Simulated CRYSTAF profile of an olefin copolymer made on a five-site-type catalyst. The dotted line indicates the parent Stockmayer’s distribution for the third site type. The bold upper curve is the weighted average of the individual distributions per site type. Simulation parameters are: TI = 1.3 x 10-4, FI1 = 0.95, ml = 0.16/t2 = 4.0 x lo4, Flz = 0.93, mz = 0.391~~ = 1.25 x it’, F13 = 0 . 8 5 , m 3 = 0 . 1 6 / t ~ = 4 . 2 x l O ~ 3 , ~ . 1 4 = 0 . 8 0 , m 4 = 0 . 1 7 / t ~ = 3 . 0 x 1 ~ 2 , ~ l ~ = 0 . 7 5 , m s = 0 . 1 2 . F o r a l l site types, r1r2 = 1 .O

7.00E-01 7.5OE-0 I 8.OOE-01 8.5OE-01 9.00E-01 9.5OE-01 I.OOE+OO

Molar fraction of ethylene (F, )

Fig. 12. Effect of asymmetric peak broadening on location of peak maximum (7 =1.25 x l t 3 , Fl = 0.85, rlr2 = 1.0,d = 0.00037)

1926 J. B. P. Soares, B. Monrabal, J. Nieto, J. Blanco

age” parameters found in the range of r, = 500-900 were used for this simulation. It is clearly shown how the CCDs of polymer chains made on sites with different copolymerization reactivity ratios will merge as a result of peak broadening during fractionation, generating the typical broad and bimodal CRYSTAF and TREF detector responses of linear low density polyethylene.

An interesting consequence of asymmetrical peak broadening in CRYSTAF and TREF is illustrated in Fig. 12. The peak maximum shifts from its actual position as the value of the parameter A3 increases. Since the posi- tion of the peak maximum is generally used for calibra- tion of both CRYSTAF and TREF, this phenomenon can lead to small distortions in the obtained calibration curve.

A more challenging application of the methodology proposed herein is to recover the actual CCDs per site type from the knowledge of CRYSTAF (or TREF) profile of a polyolefin made on a multiple-site-type catalyst. This will be the subject of a future publication.

Conclusions CRYSTAF results confirm the theoretical predictions of Stockmayer ’s distribution: the breadth of the chemical composition distribution increases with decreasing num- ber average molecular weight for polymers made with single-site-type catalysts.

Instrumental peak broadening in CRYSTAF analysis of single-site-type polyolefins was investigated with the help of Stockmayer’s distribution and a generic instru- mental spreading function. Asymmetrical peak broaden- ing seems to take place during the fractionation. It is cer- tain that similar results would be also obtained for TREE The methodology proposed herein permits the quantifica- tion of this phenomena during polyolefin fractionation.

The extension of these methodology for polyolefins made with multiple-site-type catalysts is straightforward and might lead to a better understanding of the active sites present on these catalytic systems.

Appendix The following functional dependencies for o2 and A3 on ethylene mole fraction in the general spreading function defined by Eqs. (8) and (9) were attemped:

A3 = A : + Oexp( l/Fl) (A.4)

A3 = A: + Oexp( l/F,)”’ ( A 4

Eqs. (A.1) to (AS) were used alone or in combination, for instance Eq. (A.l) and (A.4) to simultaneously account for non-uniform broadening due to o and A3 vari- ations. None of these equations provided a better fit than the one obtained with Eq. (1 1).

’) J. B. P. Soares, A. E. Hamielec, Macromol. Theory Simul. 4, 305 (1995)

2, J. B. P. Soares, A. E. Hamielec, “Temperature rising elution fractionation ”, in “Experimental Methods in Polymer Char- acterization”, R. A. Pethrick, Ed., John Wiley & Sons, in press

3, F. Defoor, G. Groeninckx, P. Schouterdan, B. van der Heij- den, Polymer 33,3878 (1992)

4, B. Monrabal, J. Appl. Polym. Sci. 52,491 (1994) 5, A. E. Hamielec, J. B. P. Soares, Prog. Polym. Sci. 21, 651

6, W. Kaminsky, Macromol. Chem. Phys. 197,3907 (1996) ’) J. B. P. Soares, J. D. Kim, G. L. Rempel, Znd. Eng. Chem.

Res. 36, 1144 (1997) 8, L. D’Agnillo, J. B. P. Soares, A. Penlidis, J. Polym. Sci., Part

A: Polym. Chem. 36,83 1 (1 998) 9, L. D’Agnillo, J. B. P. Soares, A. Penlidis, Macromol. Chem.

Phys. 199,955 (1998) lo) B. Monrabal, J. Blanco, J. Nieto, J. Soares, “Characteriza-

tion of homogeneous ethylene/l -octene copolymers made with a single-site catalyst: CRYSTAF analysis and calibra- tion ”, J. Polym. Sci., Part B: Polym. Phys., in press

( 1996)

11) W. H. Stockmayer, J. Chem. Phys. 13, 199 (1945) 12) A. Penlidis, A. E. Hamielec, J. F. MacGregor, J. Liq. Chro-

13) T. Provder, E. M. Rosen, Sep. Sci. 5,437 (1970) 14) T. Provder, E. M. Rosen, Sep. Sci. 5,485 (1970) 15) W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery,

“Numerical Recipes in FORTRAN - The Art of Scientific Computing”, Cambridge University Press, Cambridge, 2”d edition 1992

16) L. H. Tung, F. R. Runyon, J. Appl. Polym. Sci. 13, 2397 (1969)

matogl: 6, 179 (1983)