Rheological study on mixtures of different molecular weight hyaluronates

Rheological study on mixtures of different molecular weight hyaluronates N. B e r r i a u d , M . M i l a s * a n d M . R i n a u d o Centre de Recherches sur les Macromol6cules Vdg6tales (CERMAV), affiliated with the Joseph Fourier University Grenoble, BP 53 X, 38041 Grenoble Cedex 9, France Received 2 December 1993; revised 8 February 1993

The shear flow of hyaluronate solutions prepared from mixtures of hyaluronates with different molecular weights has been studied. This paper shows that prediction of the rheological behaviour of polymer mixtures at a given polymer concentration is possible knowing the intrinsic viscosity of the different mixed samples and the general variation of the specific viscosity versus the overlap parameter C[~/]. These characteristics depend on the total ionic strength in solution and have been determined for three conditions: in 0.1 M NaCI and at ionic strengths corresponding to hyaluronate concentrations of 5 and 1 0 g I - ' in pure water. The differences between the predicted viscosities and the measured ones are usually lower than 5% for the three conditions tested.

Keywords: hyaluronates; viscosity; polydispersity

Hyaluronic acid is a water-soluble glycosaminoglycan consisting of alternating flD(1--*4)-linked 2-acetamido-2deoxy-D-glucose and flD(1--*3)-linked D-glucuronic acid. It occurs in many living substrates such as synovial fluid, the vitreous humour, rooster combs and the umbilical cord. Indications of pharmacological activity have been reported in the treatment of inflammatory and degenerative joint diseases L2, and hyaluronate is used as a replacement for the vitreous fluid during ophthalmic surgery 3. The increasing interest in the industrial production of hyaluronate is due to both its potential cosmetic applications 4 and the possibility of producing bacterial hyaluronate from Streptococcus zooepidemicus. The rheological properties of hyaluronate solutions are important in these applications. These properties depend on the hyaluronate molecular weight 5 and, as hyaluronates are charged polymers, on the total ionic strength of the solution 6. Unfortunately, it is often difficult to obtain bacterial hyaluronates with the same characteristics from different fermentations, and consequently it is difficult to achieve a reproducible solution viscosity. It would be useful to be able to prepare solutions with constant viscosity, at given conditions and polymer concentrations, from mixtures of different molecular weight hyaluronates. For this purpose, this paper tests the additivity rules applied to intrinsic viscosities and molecular weights and consequently to the rheology of hyaluronate mixtures, knowing that a similar approach has recently been described for predicting the viscosity of non-interacting mixed polysaccharide solutions based on coil occupancy 7. * To whom correspondenceshould be addressed Ol41-8130/94/030137-06 © 1994Butterworth-HeinemannLimited

It occurs in many living substrates such as synovial fluid, the vitreous humour, rooster combs and the umbilical Experimental The study was carried out on five commercial bacterial sodium hyaluronates (HA-1 to HA-5) kindly given by ARD (Agroindustrie Recherches et Developpements) (Pomacle, France). The characteristics of the samples are given in Table 1. The moisture content was determined by thermogravimetry before the experiments were carried out using a thermobalance (Setaram, France, model G70). Solutions of different polymer concentrations were prepared by dissolving the samples in distilled water or in 0.1 M NaCI or by dilution of a stock solution (1 g1-1) with the corresponding solvent. For each concentration, mixtures of two different hyaluronates were prepared. The characterization of the polymers, e.g. determination of the molecular weight distribution, the average molecular weight and the intrinsic viscosity, was carried out at 29°C on hyaluronate solutions (0.75 g 1-1) in 0.1 M NH4NO3 by steric exclusion chromatography (SEC), using the multidetection equipment described previously8 and user-written software.The polydispersity of the different samples was -~w/Mn '~ 1.3. The refractive index increment measured at 25°C with a differential refractometer (Brice Phoenix, USA) was 0.155 ml g-1 in 0.1 M NH4NO3. Viscosity measurements were performed at 25 + 0.01 °C using a Contraves Low Shear 30 viscometer (Contraves, Switzerland) at a range of shear rates from 10 -2 s -1 to 128s -1 and a CS 50 rheometer (Carri-Med, UK) equipped with a Rheo 1000 C system and 5.0 software (Carri-Med) which allows direct viscosity shear rate

Int. J. Biol. Macromol. Volume 16 Number 3 1994 137

Rheological behaviour of hyaluronates: N. Berriaud et al. Table 1 Characteristicsof the hyaluronate samples Hyaluronate HA-I HA-2 HA-3 HA-4 HA-5

[~]"

(mlg-l) 1434 2375 1934 2545 2395

k= 0.42 0.43 0.39 0.38 0.43

In7

(mlg-l) 3567 5968 -

[~]:

(mlg-l) 2684 4382 3444 4462

~x10-6 0.861 1.435 1.15 1.6 1.4

B0 L=.^ A^

.

o. E

1

HA-I

•

25175

"~C

50150

XN=

75125

o= Do

02

• • oa o o

"In 0.1M NaC1 bFor Cr=8.7 x 10-Smol1-1 eFor CT= 17.5x 10-s moll -~ determination by angular speed control. The different geometries used were cones (4 and 6 cm diameters, 1 and 4 ° cone angles). The shear stress varies from 0.005 to 300 Nm -2 depending on the geometry used, and the approximate shear rate varies from 10 -1 to 2000s -1 depending on the experimental conditions.

o

•

o

• •

o

•

o

o

ao s eo o o ee

• o •

•

o

1 01

........ 10 "1

i

I

i

I illlll

i

101

100

l

I l Illll

l

l

l ' lllll

l

1 os

102 -~ (s -1)

I

IIIIII

1o4

Figure I Shearrate dependenceof viscosityfor HA-I, HA-2 and their mixtures. Polymer concentration 5 g1-1 in 0.1 M NaCI at 25°C

Results and discussion Different representations have been described in the literature to relate the molecular weight contribution to the viscosity. One of the most frequently used involves plotting viscosity change as a function of the overlap parameter CD/], where C is the polymer concentration (g ml-1) and [~/] is the intrinsic viscosity (ml g-1). This representation is expected to hold for low and moderate polymer concentrations. At higher concentrations, the parameter C[r/] can be favourably replaced by C.M 9A0. In fact, these representations are average descriptions which do not take into account the difference in behaviour in semi-dilute and concentrated regimes s'1°'~1. At first, this study was carried out in 0.1 M NaCI to limit electroviscous effects. Then it was repeated at total ionic concentrations CT corresponding to 5 and 10g1-1 hyaluronate solutions in water. This last condition was chosen because it corresponds to a classic condition for marketing and some uses of hyaluronate.

Rheological study of hyaluronate mixture in O. 1 M NaCl Effect of shear rate. Polymer solutions exhibit reversible pseudoplastic behaviour. An example of a doublelogarithmic plot of the viscosity ~/versus the shear rate (~) for HA-I, HA-2 and their mixtures at 5 g 1-1 is shown in Figure 1. These curves can be characterized by three parameters: the Newtonian viscosity (r/o), the critical shear rate corresponding to the onset of the shear thinning behaviour (~c) and the slope (s) of the curve in the shear-dependent viscosity domain. Above $¢, log r/decreases linearly with log ~)~according to the power law ~/= kS5. The origin of this viscosity decrease may be due to the increase of molecular orientation in the flow, to the reduction in the extent of entanglement coupling and to intramolecular hydrodynamic interactions. Figure 2 shows the slope variation (pseudoplasticity) versus the overlap parameter C[r/]. The theoretical limit is - 0 . 8 1 8 for polymers 1°. This limit is generally obtained when C[~/] is higher than 40. The hyaluronate mixtures show no difference compared with separated polymers and the classic behaviour of polymer in solution is observed ~o. Intrinsic viscosities. The viscosities in the Newtonian region allow us to determine the intrinsic viscosity [r/'l from extrapolation of the specific viscosity to zero

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Int. J. Biol. Macromol. Volume 16 Number 3 1994

0.8 ~

y

.-.----'-'-7~-. ~I a

o

0.6

y

I 0.4

~/ ~o o•

0.2

o

HA-1

•

25/75

• o

50/50 75/25

A 0

~ 0

I 10

~

I

~

20

I 30

~

HA-2 I 40

50

c [~1]

Figure 2 Slope(s) of the power law log r/vs. log ~' plotted against the overlap parameter for HA-I, HA-2 and their mixtures

concentration for the hyaluronates and their mixtures. The intrinsic viscosity of the hyaluronate mixtures can be given by the additivity relation: ['/7"] = ~ ' ~ O ) i [ t l i ' ]

(1)

where i represents the hyaluronate species with intrinsic viscosity [r/i] and coi is the mass fraction of species i in the mixture. Experimental and calculated values of intrinsic viscosities are reported in Table 2 for samples HA-1 and HA-2. There is good agreement between these values, so it is possible to predict by additivity the intrinsic viscosities of mixtures in 0.1 M NaCl.

Molecular weights. The molecular weights of the hyaluronates and their mixtures were determined by gel permeation chromatography in 0.1 M N H 4 N O 3. The average molecular weight of the mixtures can be calculated from the relation: Experimental and calculated values are compared in

Table 3 for samples HA-1 and HA-2. There is good agreement between the predictions and the experimental measurements. No differentiation between the two polymers in the solutions can be detected by SEC spectra.

Rheological behaviour of hyaluronates: N. Berriaud et al.

Prediction of specific viscosities. Specific viscosities in the N e w t o n i a n p l a t e a u were p l o t t e d as a function of the o v e r l a p p a r a m e t e r C[r/] (Figure 3). T h e curve presents two a p p r o x i m a t e l y linear parts: below C* a n d for the largest p o l y m e r c o n c e n t r a t i o n s . C* is the critical

c o n c e n t r a t i o n , a p p r o x i m a t e l y equal to [~/]-1, a b o v e which p o l y e l e c t r o l y t e chains are entangled. Below C* (C*[r/] ~ 1), the H u g g i n s law r/,p= C[r/] + k , ( C [ r / ] ) 2 is verified, a n d ~/is p r o p o r t i o n a l to M as found recently for h y a l u r o n a t e solutions 5. This is the dilute regime. In the r e p r e s e n t a t i o n given in Figure 3 for m o r e c o n c e n t r a t e d solutions, l o c a t i o n of the semi-dilute regime, which d e p e n d s on the m o l e c u l a r weight, is difficult 5. Thus, Figure 3 represents the average b e h a v i o u r of the different m o l e c u l a r weight s a m p l e s in semi-dilute a n d c o n c e n t r a t e d regimes where the average slope of the straight line is 4.18 a n d t / i s p r o p o r t i o n a l to M 4 as found previously 5. The curve in Figure 3 can be c h a r a c t e r i z e d in a first a p p r o x i m a t i o n , in the d o m a i n of c o n c e n t r a t i o n s a n d m o l e c u l a r weights tested, by the relation:

Table 2 Comparison of experimental and calculated intrinsic viscosities for different mixtures of hyaluronate HA-I and HA-2 in 0.1 M NaCI at 25°C

[n]...

[~]=.,

HA-2/HA-1 ratio

(ml g- 1)

kn

{ml g- 1)

25/75 50/50 75/25

1670 1942 2162

0.427 0.396 0.415

1669 1904 2140

n,p = CEr/] + k.(CFn]) 2 + B(CEr/])"

T h e k n value c o r r e s p o n d s to the H u g g i n s constant. It is i n d e p e n d e n t of m o l e c u l a r weight. B a n d n are equal to 7.77x 10 - 3 a n d 4.18, respectively. It s h o u l d then be possible to predict the viscosity of a m i x t u r e by replacing C[r/] in E q u a t i o n 3 o r Figure 3 by ECi[r/]i. A similar m e t h o d has a l r e a d y been tested with success o n cellulose derivative mixtures 7. The e x p e r i m e n t a l values are c o m p a r e d with the c a l c u l a t e d ones in Table 4. The a c c u r a c y between these

Comparison ofexperimental and calculated molecular weights for different mixtures of hyaluronate HA-1 and HA-2 in 0.1 M NaCI at 25°C Table 3

Experimental h3,, X 10 - 6

Calculated

HA-2/HA-1 ratio 25/75 50/50 75/25

0.991 1.141 1.296

1.004 1.148 1.292

(3)

/~w x 10 - 6

Table 4 Comparison between calculated and experimental specific viscosities for different mixtures of HA-I and HA-2 in 0.1 M NaCI at 25°C

Mixture HA-2/HA-1

Concentration (g I- 1)

C[r/]

Calculated r/,v (equation (3))

Experimental t/,p

25/75 [r/] = 1669 mi g- ~

0.0961 0.241 0.384 0.48 0.721 0.961 2.415 4.83 9.7 19.3

0.160 0.402 0.642 0.802 1.203 1.61 4.03 8.061 16.19 32.2

0.171 0.47 0.82 1.08 1.83 2.76 13.5 83 1007 16073

0.1689 0.478 0.8195 1.0795 1.814 2.925 12.93 82.39 981 16855

50/50 [r/] = 1904 ml g- 1

0.097 0.241 0.386 0.48 0.724 0.967 2.427 4.85 9.707 19.38

0.185 0.461 0.736 0.914 1.379 1.841 4.621 9.234 18.482 36.9

0.199 0.55 0.966 1.27 2.21 3.36 18.26 129 1695 28189

0.2021 0.563 0.965 1.281 2.189 3.575 17.4 127.13 1636 29369

75/25 [r/] = 2140 ml g- 1

0.097 0.243 0.388 0.486 0.728 0.972 2.439 4.89 9.719 19.45

0.208 0.52 0.83 1.04 1.56 2.08 5.22 10.46 20.8 41.6

0.23 0.63 1.12 1.5 2.63 4.1 24.4 198 2714 46292

0.23 0.64 1.12 1.5 2.62 4.4 23.6 191 2662 58572

Int. J. Biol. Macromol. Volume 16 Number 3 1994

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Rheologica/ behaviour of hyaluronates: IV. Berriaud

e t al.

values is higher t h a n 5%. K n o w l e d g e of C a n d [q] allows p r e d i c t i o n of the specific viscosity of h y a l u r o n a t e solutions at least for the c o n c e n t r a t i o n s a n d m o l e c u l a r weights used in this work. In o r d e r to test o u r model, Table 5 c o m p a r e s the specific viscosities for m i x t u r e s of two o t h e r samples, H A - 3 a n d HA-4. T h e a g r e e m e n t between these values is good.

H y a l u r o n a t e is a polyelectrolyte: its p h y s i c o c h e m i c a l p r o p e r t i e s d e p e n d on the t o t a l ionic c o n c e n t r a t i o n C r which is given by: Ca-= ~ C p + C,, where Cp a n d Cs are the c o n c e n t r a t i o n s of the polyelectrolyte (expressed as mol of C O O - per litre) a n d of external salt (mol 1-1), respectively, a n d • is the o s m o t i c coefficient of the c o u n t e r i o n s in the absence of external salt ( ~ = 0 . 7 for hyaluronate). T w o different total ionic c o n c e n t r a t i o n s were chosen: C T I = 8 . 7 x 10 - 3 m o l l -1 a n d CTz = 17.5 x 1 0 - 3 m o l l -~, which c o r r e s p o n d to the ionic activities (~Cp) of 5 g 1-1 a n d 10 g 1- ~ h y a l u r o n a t e solutions in distilled water, respectively. T h e dilutions were achieved following the iso-ionic procedure: the decrease in p o l y m e r c o n c e n t r a t i o n , ACp, is c o m p e n s a t e d for by an increase in salt c o n c e n t r a t i o n , AC,, equal to ~ACp, in o r d e r to m a i n t a i n CT constant.

Rheological study of hyaluronate mixtures at low ionic strength T h e objective was to define r/sp versus (C[r/]) for h y a l u r o n a t e s o l u t i o n s at 5 a n d 10 g 1-1 in distilled water.

Table 5 Comparison between calculated and experimental specific viscosities for different mixtures of HA-3 and HA-4 in 0.1 i NaCI at 25°C Calculated Mixture HA-3/HA-4

Concentration (g I- 1)

C[r/]

(Equation(3))

Experimental q,p

59.91/40.09 46.79/53.21 16.92/83.08

4.997 7.41 10.07

10.89 16.74 24.59

229 1148 5334

196 1276 5377

r/,p

Intrinsic viscosities. The experimental intrinsic viscosities It/] were d e t e r m i n e d , using iso-ionic dilution, by extrap o l a t i o n o f the specific viscosities to zero c o n c e n t r a t i o n

105

A

HA-2 (Mw- 1 435 000)

#

II I

/ /

Table 6 Comparison of experimental and calculated intrinsic viscosities for different mixtures of hyaluronate HA-1 and HA-2 at Crt and Cr2 at 25°C [~]l exp

CT

[-~l¢lll

(tool I- 1)

HA-2/HA-1 ratio

(ml g- 1)

kn

(ml g- 1)

8.7 x 10 -3

0/100 100/0 25.2/74.8 45.9/54.1 74.8/25.2

3567 5968 4180 4681 5366

0.213 0.285 0.232 0.265 0.262

4172 4669 5363

17.5 x 10-3

0/100 100/0 25.2/74.8 45.9//54.1 74.8/25.2

2684 4382 3112 3466 3955

0.247 0.297 0.265 0.287 0.289

3112 3464 3954

101

10. 1

r

........

1 O" 1

I

.......

100

=1

~

= ~ =====

101

102

cM

Figure 3 Specific viscosity as a function of the overlap parameter (C[r/]) for HA-I and HA-2 in 0.1 M NaC1 at 25°C

Table 7 Comparison between calculated and experimental specific viscosities for different mixtures of HA-I and HA-2 at Crt = 8.7 x 10- 3 Eq 1- t and 25°C Mixture HA-2/HA-I

Concentration (g l- 1)

C[q]

Calculated q,v (equation (4))

Experimental q,p

25.2/74.8 [q] =4172 ml g- 1

0.0963 0.193 0.385 0.481 0.674 4.83

0.402 0.805 1.606 2.01 2.811 20.15

0.442 0.97 2.25 3.0 4.83 251

0.451 0.95 2.2 2.97 4.71 236

45.9/54.1 [q] = 4669 ml g- 1

0.097 0.194 0.388 0.484 0.678 4.84 0.098 0.196 0.393 0.491 0.689 4.9

0.452 0.906 1.811 2.26 3.165 22.6 0.526 1.051 2.108 2.633 3.695 26.28

0.5 I. 11 2.64 3.55 5.74 355 0.59 1.33 3.2 4.4 7.3 572

0.51 1.12 2.69 3.6 5.87 346 0.61 1.35 3.2 4.4 7.4 578

74.8/25.2 [q] = 5363 ml g- 1

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16 Number

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Rheological behaviour of hyaluronates: IV. Berriaud

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Table g Comparison between calculated and experimental specific viscosities for different mixtures of HA-1 and HA-2 at CT2 = 17.5 x 10-3 mol Iand 25°C Mixture HA-2/HA-1

Concentration (g 1- ~)

Calculated r/,p (equation (5))

25.2/74.8 [7] =3112ml g - t

0.098 0.193

0.304 0.6

0.33 0.7

0.33 0.69

0.387

1.204

1.6

1.586

0.483

1.503

2.13

2.1

0.676

2.103

3.35

C[~/]

9.667

30.08

Experimental ~,,p

3.34

1767

3.107

1452

30.6/69.4

0.97

[7] =3204mlg -~

1.94 4.37

45.9/54.1

0.097

0.336

0.37

0.37

[7] =3464ml g - i

0.194 0.388

0.671 1.344

0.79 1.84

0.8 1.85

0.484

1.676

2.5

2.5

0.678

2.348

3.92

4.03

6.22 14

9.687

33.55

5.9

5.7

19.8 142

18.6 147

2625

2871

74.8/25.2

0.098

0.387

0.43

0.43

[7] =3954mlg -I

0.196 0.393

0.775 1.554

0.94 2.23

0.94 2.24

0.491

1.941

3

3.06

0.689

2.72

4.87

4.96

9.733

38.5

4348

Table 9 Comparison between calculated and experimental specific

1 03

viscosities for different mixtures of HA-3 and HA-5 at CT = 17.5 x 10-3 mol 1- x and 25°C Mixture HA-3/I-IA-5

Concentration (g I- x)

100/0 0/100 75/25 33.4/66.6 25/75 22.6/77.4

10 10 10 10 10 10

C[~/]

Calculated ~/sp (equation (5))

Experimental r/,p

34.44 44.62 36.98 37.82 42.1 42.32

2889 7558 3753 4078 6073 6198

2895 7557 3379 3990 6087 6222

3929

,o, I::"1 j// 1° 1

,oo j \ %p - C N + 0 249x(C 1111)2 + 8.37 x 1O"4xlC [rd)398

10-1

for HA-I, HA-2 and their mixtures. Intrinsic viscosities of the mixtures were calculated from equation (1). Experimental and calculated values are reported in Table 6. There is good agreement. Therefore, it is also possible to predict the intrinsic viscosities of mixtures at CT~ and CT2 (low ionic strengths).

Prediction of specific viscosities. The specific viscosities in the Newtonian plateau for HA-1 and HA-2 were plotted as a function of the overlap parameter C[r/] for Crl (Figure 4) and CT2 (Fi#ure 5). The curves can be characterized in a first approximation for this domain of concentrations and molecular weights by equation (3) to give: rhp = C[r/] + 0.249(C[r/]) 2 + 8.37 x 10- 4"(C[t]])3"98

(4)

a(C[~])3"91

(5)

~/sp = C[r/] + 0.272(C[r/]) 2 + 2.46 x I0-

........

I

1o-I

........

1o0

I

........

~o~

lO 2

c [n]

Figure 4 Specific viscosity as a function of the overlap parameter (C[~/]) for HA-1 and HA-2, Cr=8.7 x 10 -3 mol I- ~ at 25°C 1 04

1 03

I1 z.

//

HA-2

102 ¢,-

1 01

10 °

for CT~ and CT2, respectively. Knowledge of C and [r/'] allows prediction of the specific viscosity of hyaluronate solutions for at least the concentrations and molecular weights in the range used. As [r/] for mixtures of hyaluronate can be estimated from the intrinsic viscosity

IlSp -C[qJ +0.272x(C{rlJ)

10-1

~ 1 0" 1

.......

I 1 00

........

2

I

3 3 91 +2.457x10" x(C{~) "

........

1 01

! 1 02

c [~] Figure 5 Specific viscosity as a function of the overlap parameter (C[r/]) for HA-I and HA-2, CT= 17.5 X 10 -3 mol I- t at 25°C

Int. J. Biol. Macromol. Volume 16 Number 3 1 9 9 4

141

Rheological behaviour of hyaluronates." IV. Berriaud et al. of separate samples, obtained at the same Cr, it should be possible to predict from equation (1) the specific viscosities of hyaluronate mixtures in these conditions. Comparisons between experimental and calculated values for the four hyaluronate samples used are given in Tables 7, 8 and 9. The agreement between the calculated and experimental values is good and differences lower than 5% are usually observed.

Conclusion Mixtures of two different molecular weight hyaluronates in solution exhibit rheological behaviour equivalent to that of a pure hyaluronate sample. The theoretical intrinsic viscosity of hyaluronate mixtures can be estimated from the classic additivity law which applies to hyaluronate. At low ionic content, the iso-ionic dilution must be used to measure the intrinsic viscosity of the separated samples and then to predict the intrinsic viscosity of the mixtures. As the viscosity, at given ionic strength and temperature, is a function in a first approximation of the overlap parameter C[~/] even for hyaluronate mixtures, it is possible to predict with good accuracy the solution viscosity of mixtures for a definite polymer concentration whatever the total ionic content. A similar approach has already been tested to study polysaccharide interactions 7.

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Acknowledgements The authors are grateful to F. Launay and R. de Baynast from the Laboratories ARD (Pomacle, France) for financial support, scientific cooperation and donation of samples.

References 1 Asheim,A. and Lindbald, G. Acta Vet. Scand. 1976, 17, 379 2 Numiki,O., Toyoshima, M. and Morisaki, N. Clin. Orthop. 1982, 146, 260 3 Balazs,E.A. in 'Healon: A Guide to its Use in Ophthalmic Surgery' (Eds D. Miller and R. Stegmann), Wiley, New York, 1983, p 5 4 Balazs, E.A. and Band, P. Cosmetics and Toiletries 1984, 99, 65 5 Fouissac, E.. Milas, M. and Rinaudo, M. Macromolecules 1993, 26, 6945 6 Fouissac,E., Milas, M., Rinaudo, M. and Borsali,R. Macromolecules 1992, 25, 5613 7 Manniou, R.O., Mitchell, J.R., Harding, S.E., Lee, N.E. and Melia, C.D. in 'Gums and Stabilisers for the Food Industry' (Eds G.O. Phillips, P.A. Williams and DJ. Wedlock), IRL Press, Oxford, 1990, volume 5, p 451 8 Tinland, B., Mazet, J. and Rinaudo, M. Makromol. Chem. Rapid Commun. 1988, 9, 69 9 Ferry,J.D. in 'Viscoelasticproperties of polymers',3rd edn, Wiley, New York, 1980, chapter 17, p 486 10 Graessley, W.W. Adv. Polym. Sci. 1974, 16, 1 11 Takahashi,Y., Isono,Y., Noda, I. and Nagasawa,M. Macromolecules 1985, 18, 1002