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Viscosity of semi-dilute polymer solutionsM. Adam, M. Delsanti

To cite this version:M. Adam, M. Delsanti. Viscosity of semi-dilute polymer solutions. Journal de Physique, 1982, 43 (3),pp.549-557. �10.1051/jphys:01982004303054900�. �jpa-00209425�

549

Viscosity of semi-dilute polymer solutions

M. Adam and M. Delsanti

Laboratoire Léon-Brillouin, CEN Saclay, 91191 Gif sur Yvette Cedex, France

(Reçu le 25 mai 1981, révisé le 24 aofit, accepté le 3 novembre 1981)

Résumé. 2014 Nous présentons des résultats de viscosité obtenus dans des solutions semi-diluées de polymères(c* c 10%).La variation de la viscosité est indépendante de la température de transition vitreuse du polymère et compatibleavec un coefficient de frottement du monomère proportionnel à la viscosité du solvant.En fonction de la concentration dans différents systèmes, les relations suivantes sont obtenues :2014 PIB-toluene, ~r ~ c4,1, qui est proche de la théorie de la viscosité de de Gennes,2014 PS-benzène et PIB-cyclohexane, ~r ~ c4,6, qui peut être expliquée par la structure locale de la chaîne,2014 solution semi-diluée à la température 03B8, ~r ~ c5, qui peut être interprétée avec un temps de reptation analogueà celui introduit dans le modèle de de Gennes et une élasticité déduite de la théorie de champ moyen.En fonction de la masse moléculaire, nous obtenons :

~r ~ M3,1wen accord avec le modèle de reptation.Toutefois, il semble que les lois d’échelle en température ne peuvent expliquer la décroissance de la viscosité rela-tive lorsque la température croit.

Abstract. 2014 We report viscosity measurements on semi-dilute solutions (c* c 10 %).The viscosity variation is independent of the glass transition temperature of the undiluted polymer and consistentwith monomer friction proportional to the solvent viscosity.With concentration, the following variations were observed :2014 for PIB-toluene, ~r ~ c4.1, close to de Gennes’ viscosity theory,2014 for PS-benzene and PIB-cyclohexane, ~r ~ c4.6 which seems to be influenced by local statistics in the chains,2014 for semi-dilute 03B8 solvent conditions, ~r ~ c5 which could be interpreted using de Gennes’ theory of reptationand mean field theory of elasticity.With molecular weight we observe :

~r ~ M3.1win agreement with reptation model.However temperature scaling laws do not seem to be applicable. The interpretation of the temperature depen-dence of the viscosity remains open question.

J. Physique 43 (1982) 549-557 MARS 1982,

Classification

Physics Abstracts46.30J - 51.20 - 61.40K - 82.90

Introduction. - Zero shear viscosity of polymersolutions and melts have been intensively studied formany years. A good review article on the subject waswritten by Fox and Berry [1] (1968).Our purpose is to compare de Gennes’ viscosity

theory [2], which gives a new approach on the problem,to experimental results obtained with semi-dilutesolutions. A semi-dilute solution may be consideredas a transient network, the monomer concentration c

being higher than the overlap concentration c*

(c* = N/R3 [3]) but much lower than the solventconcentration. In order to be able to compare theory

with experiments we have performed some newexperiments on very high molecular weight polymer.

First, we review de Gennes’ theory of semi-dilute(S.D.) polymer solutions. Then we define the experi-mental conditions at which the experiments wereperformed. Finally, we present and discuss the expe-rimental results obtained in both, good and 0 solvents.

1. Theory. - The main term in the expression forthe viscosity 11 is the product of the longest relaxationtime TR and the shear elastic modulus E of the tran-sient network :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004303054900

550

Let us successively derive the two quantities Eand TR from de Gennes’ theory in the simplest caseof an athermal solvent (Flory-Huggins parameterz = 0).The main hypothesis of the de Gennes’ theory is

to consider a chain (in a S.D. solution) as formed bya succession of subchains of mean size ç, each sub-chain having g monomers.

Inside a subchain, the g monomers are subject toexcluded volume effects and to hydrodynamic inter-actions. Such interactions do not take place betweentwo subchains; ç is a screening length for both exclud-ed volume and hydrodynamic effects.

Edwards [4] and de Gennes [5] imagine that themotion of a particular chain is a reptation in a ficti-tious tube, defined by all the surrounded chains. Thediameter of this tube is the screening length ç :

where I is an effective length per monomerThe length of the tube is :

where N is the number of monomers in a chain and

since a S.D. solution is a closely packed system ofsubchains units.The time TR necessary for a complete renewal of

the tube is the time required for chain to reptate thelength of the tube [6] :

where Dt is the diffusion coefficient of the Rousechain along its tube. Thus :

where qo is the solvent viscosity.Combining equations (3), (5) and (6), we have :

thus :

Substituting the expressions (4) and (2) for g and çin the expression of TR (7) we have :

Following rubber elasticity theory (7), the elasticmodulus of the transient network is proportional tothe volume number density of strands c/g. Using theexpressions (2) and (4) we obtain : E -- kT/ç3 thus :

From the expressions of T R (8) and E (9) we obtainfor the viscosity [10] :

or

Using the most accurate experimental value of v[8], which corresponds to the prediction of the nvector model [9],

one finds :

We have considered above the case of an athermalsolvent (x = 0, v = 0.588) where only one charac-teristic length ç exists. The quality of the solventshould have no influence on the viscosity molecularweight exponent, but it must influence strongly theconcentration dependence. One may remark that,since the length ç decreases when the quality of thesolvent increases (Ref [2], p. 119, 1 l, 12), the viscositygiven by equation (10) must increase.

In the case of a 0 solvent [13] the theory of the vis-cosity is much more difficult because there exist twolengths [14] :- the correlation length, which is the screening

length of the density-density correlation function :

- and the mean distance between two consecu-tive entanglements :

The problem is : how the shear modulus and thelongest relaxation time depend on ç and Ç2. In sec-tion 3.2, we will discuss our experimental resultsobtained in 0 conditions and compared them withdifferent theoretical possibilities.

2. Experimental procedure. - The principle of theviscometer was given in our earlier paper [15]. Wemeasure the viscous force F to which a sphere (radiusr = 5 x 10-2 cm) is submitted when the fluid is

displaced at a velocity v :

551

Table I. - Characteristics of polymer solutions studied.Tg is the glass transition temperature of the undilutedpolymer and X the Flory-Huggins parameter. Indexesv and t refere to X deduced from intrinsic viscosity andthermodynamic measurements, respectively.

we measure the ratio F/v as a function of v and verifythat F/v has a constant value, thus we determine thezero shear viscosity of the sample.We have made various improvements [16] upon

the viscometer specifications as compared to refe-rence [15]. The force maintaining the sphere in placeis transmitted magnetically and we have improved thefeedback control of the current driving the fieldcoils. This current is of the order of 1 mA and it isnow measured to within 1 %. The sample flow is

Fig. 1. - Variation of the overlap concentration c* as afunction of molecular weight M.. + - + PIB and .--8PS samples on which concentration exponent values aredetermined. The numbers on each line represent the Mw/Mnvalues. a) Good solvent systems. - PS-benzene c* values;.... PIB-cyclohexane c* values; ------ cc deducedfrom [27]. b) 6 solvent systems. - PS-cyclohexane (35 OC);· · · · PIB-benzene (25 °Q ; ------ Cc

now driven by a stepmotor and its velocity is measur-ed to an accuracy better than 0.1 %.We made an absolute calibration of the apparatus

by measuring a standard silicon oil (q = 50 P at25 OC) and our new overall precision is 2 %.From the measured viscosity, we have deduced the

relative viscosity defined as the ratio of polymer solu-tion viscosity to that of the solvent [17], given in appen-dix 2.Two kinds of polymer were used : poly-isobutylene

and polystyrene in three different solvents whosecharacteristics are given in table I.

In figure 1 we plot the different concentrations ofthe samples at which the viscosity was measured. Werepresent also the variation of the overlap concentra-tion c* as a function of molecular weight. As usual [12,15,18] we use for c* :

where A is Avogadro’s number, p is the density of thesolvent, and RG is the radius of gyration measured bylight scattering [19]. For comparison, we plot (dashedline) the value of cc, for PS system :

where cc (g/g) define the boundary between the entangl-ed and non entangled regime [20]. We can note that :- the cc, ~ M-1, in good solvent cannot be justifi-

ed using de Gennes’ theory,- for PS good solvent (Fig. la), with our range

of molecular weight, 1.5 cclc* 3,- for PS-0 solvent (Fig. 1 b), cc is smaller than c*

and 1.3 c*lcc 6.

Thus for such high molecular weight polymers(M > 106), we find no significant range of concen-tration where the system is semi-dilute but not

entangled [20].

3. Experimental results and discussions. - 3 .1GOOD SOLVENT. - 3.1.1 Concentration dependence.- In the PS-benzene and PIB-cyclohexane systems,the dependence of relative viscosity on concentrationis, at 32 OC, identical for both systems :

For PIB-cyclohexane the exponent decreases sli-

ghtly as the temperature increases, at 64 OC, the expo-nent value is 4.4 + 0.1. This has to be checked onPS-benzene at high temperature.For PIB-toluene the viscosity, for a given concen-

tration, is much lower than for PIB-cyclohexane :

552

Fig. 2. - Relative viscosity at 32°C as a function of con-centration in 0 PS-benzene samples (MW = 24 x 106)and + PIB-cyclohexane samples (MW =1.17 x 106) [41].

In the PIB-toluene system, the concentration depen-dence of the relative viscosity is :

in agreement with reference [21]. The concentrationexponent, close to theoretical value of 3.95, is inde-

pendent of the temperature.

Fig. 3. - Relative viscosity at 64 °C in PIB-toluene sam-ples (M, = 1.17 x 106) as a function of concentration [41].

3.1.2 Temperature dependence. - The main effect,observed on S.D. solutions, PS-benzene and PIB-cyclohexane, is that the relative viscosity q, decreasesas we increase the temperature.An example of the decrease of the viscosity as a

function of temperature is given in figure 4. Thisvariation can be considered as linear and parame-terizing it as :

we find : F = 9 x 10- 3 in the case of figure 4.

Fig. 4. - Variation of the relative viscosity as func-tion of temperature for a PIB-cyclohexane sample(c = 9.3 x 10- 2 g/g).

In figure 5 we present the variation of r as a func-tion of concentration for different systems (somenumerical values of q, as a function of temperatureare given in appendix 3).

In a dilute regime with short chains at high concen-tration (c = 9 x 10 -2 g/g, c/c* = 0.6, MW = 4 x 104)the relative viscosity is very small and within experi-mental accuracy, independent of the temperature(20 °C T 60 OC), see figure 5. Even with 9 %solution concentration, the viscosity dependence ontemperature is that of the solvent. This proves thatfor a monomer concentration less than 10 %, themonomer friction is proportional to the viscosity ofthe solvent (monomer-monomer friction can be

neglected).In figure 5, one observes that :- PS-benzene and PIB-cyclohexane systems pre-

sent the same variation of r with concentration,

Fig. 5. - Variation of r (see text) as a function of theconcentration for different good systems. + PIB-cyclo-hexane ; * PIB-toluene; 0 PS-benzene, M, = 7.1 x 106 ;A PS-benzene, MW =2 x l(f; 00 PS-benzene, Mv =4 x 104,c

C - 0.6.

553

r being independent of the molecular weight. Thistemperature behaviour is not a glass transition

temperature effect, because one set of experiments(on PS) was performed below, the other (on PIB)above the Tg of the respective undiluted polymer,- the PIB-toluene system presents a variation of

the relative viscosity with temperature which is

independent of concentration. The r values are verysmall compared to those obtained on PIB-cyclo-hexane.

In the systems studied (x = 0.44) the concentra-tion exponent values found are larger than the theo-retical value developed in section 1 for an athermalsolvent. In the systems PS-benzene and PIB-cyclo-hexane, the relative viscosity decreases as we increasethe temperature.

If the first result can be interpreted as due to highx values, the second result cannot be interpretedusing usual temperature scaling laws [2, 11, 12]which predict a thermal effect opposite to that observ-ed. Indeed increasing the temperature, the quality ofthe solvent increases and thus the viscosity shouldincrease (see section 1).

3.1. 3 Qualitative interpretation of the results. -In the polymeric systems studied (x = 0.44) theeffective length I is much larger than the size of amonomer a. Let us suppose that :

i) at very short distance r (r ma), the chain is

rigid [22, 23], m being the number of monomersaffected by rigidity,

ii) at larger distance (r > ma) the subchain ç has aGaussian conformation if m g gt. The excludedvolume effects are dominant if g > g, [24].

In the range of molecular weight studied, we havelow values of glg, in the semi-dilute range. For instance,in a PS-benzene solution, at room temperature, gtcorresponds to a molecular weight M. of = 104 [25].By comparison of diffusion coefficient at zero concen-tration and cooperative diffusion coefficient in semi-dilute solution [18], having a monomer concentrationof 2 % c 10 % the corresponding Mg values are :

Thus, for c = 10 %, the ratio g/gt is of the order ofunity. In other words, at room temperature and for2 % c 10 %, the effective value of the exponent vhas not reached its asymptotic value.

Using the assumptions i) and ii) the expressionsglgt (A. 5) and r¡r (A. 6) are developed in appendix 1.In those expressions T is a constant because the systemsstudied (PS-benzene and PIB-cyclohexane) have verylow 0 temperatures [7] and small change near roomtemperature does not affect the quality of solvents(T = const.). If we suppose that m is inversely pro-portional to the temperature we obtain the followingtemperature dependences :

When T increases, glgt and 1, increase and decrease,respectively. This seems to justify that in the case ofPS-benzene and PIB-cyclohexane :- F is independent of the molecular weight (see

Fig. 5),- the viscosity and the experimental concentration

exponent decrease as T increases.

In the case of PIB-toluene the independence of therelative viscosity with temperature and the low valueof concentration exponent could be due to highflexibility of the polymer in this system giving a highvalue of g/gt.

This agreement is only qualitative, we cannot gofurther because the effective v is an unknown functionof concentration and temperature.

This assumption (that v has not reached its asymp-totic value) seems in contradiction with neutron

measurements on ç in PS-benzene solutions [26].But one must note that the law

in which the exponent value corresponds to the

asymptotic value of v, was measured in a concentra-tion range (0.5 % c 5 %) lower than that of ourviscosity measurements, hence in a higher range ofgIg".

Self-diffusion measurements on PS-benzene solu-tions are reported in reference [27] (2 % c 20 Y.),they give

Using the relation : Ds L-- R 2/ TR with R 2~ C- 1/4and E - c2.2s (corresponding to v = 0.6) leads to

q - C3.7. This exponent (3.7), in agreement withde Gennes’ prediction, is much lower than our 4.6concentration exponent. Is the concentration exponentof the self diffusion coefficient very sensitive to theeffective value of v [28] ?Due to the low value of g/gr the PS-benzene system

might be used as an experimental model of mean fieldtheory. However this theory leads to a viscosityconcentration exponent of 3.5, smaller than the goodsolvent exponent (3.95) and thus in contradiction withour experimental results.

In short, the value of the concentration exponentmay, be interpreted as due to a non athermal solventpolymer system. However, the decrease of the relativeviscosity with increasing temperature could not beunderstood in terms of the classical temperaturedependence of ç, but in terms of local rigidity.

3.2 0 SOLVENT : : PS-CYCLOHEXANE, PIB-BENZENESOLUTIONS. - The viscosity of S.D. solutions, at

0 temperature and at a given concentration is not

very different from that of S.D. solutions in goodsolvents. This contrasts with what is observed in dilute

solutions, (Ref [29]). On both our systems, when weincrease the temperature from the 0 point, (T 2013 0 = 160)the relative viscosity, for a 9 % monomer concentra-tion, decreases by a factor 1.6.

554

Fig. 6. - Relative viscosity as a function of concentrationat 6 temperature 0 PS-cyclohexane (M,, = 7 x 106),T = 35 °C, + PIB-benzene (M,,, = 1.17 x 106 ), T =25 °C[41], the arrows represent c* values.

In both cases, PS-cyclohexane (35 °C) and PIB-benzene (25°C) the relative viscosity varies withconcentration as :

In figure 6 the arrows represent the c* values on bothsystems. As soon as c > c*, the experimental points,on log log scale, lie on a straight line of slope 5.

Starting from these experimental results (1,, - c5)we want to examine the contribution of the two charac-teristic lengths which are present in the system, at the 0temperature.The first length is the mean distance between two

consecutive entanglements or contacts :

using this distance it was found [14, 30] :

The second length is the correlation length ofdensity-density correlation function :

ç ’" c- I corresponds to the expression (2) of jwhere v is set equal to 1/2. Using v = 1/2 in the expres-sions of the shear modulus (9) and of the longestrelaxation time (8) one find :

Combining the expressions of TR and E given, thedifferent resulting viscosity expressions are reportedin table II.Thus the only way to match the experimental

results is to have :

Shear modulus E - c2Longest relaxation time TR - N 3 C3 .

Physically : - the elastic modulus corresponds to

Table II. - Different expressions of the viscosity11 = ETR obtained combining the expressions of Eand T R ( 1 4 and 15).

probability of contact of two monomers, belongingto a Gaussian chain, because the coils are real andcannot cross each other.- The longest relaxation time (TR - N 3 cl) cor-

responds to a screening length proportional to c-’.In that case the fictitious tube has a length :

because

and thus a diameter proportional to j.At the 0 temperature, they are contact points

(entanglements) which define the transient network,but which do no show up in the correlations becausethe pair interaction vanishes. In order to define acorrelation length ç in the density-density correlation ,

function, one must invoke higher order interactions.However the sign of the variation of the relative

viscosity with temperature cannot be rationalizedwith the usual temperature dependence of the charac-teristic length in S.D. solutions. Using this temperaturedependence [11, 12], we find a relative viscosity whichincreases with temperature.

3. 3 MOLECULAR WEIGHT DEPENDENCE IN PS-BEN-ZENE SOLUTIONS. - The polydispersity of PS samplesmakes the determination of the value of the molecularweight exponent very delicate. We have used the bestcommercial samples available.The molecular weight exponent value is independent

of the quality of the solvent, thus of the concentrationexponent value (see section 3.1.1). Taking the experi-mental value r ’" C4.6 for PS-benzene samples at

32 OC we find :

with samples from Toyo Soda company, having apolydispersity of 1.04 and 1.14 for M, = 3.84 x 10’and Mw = 6.77 x 106, respectively; and,

with samples from pressure chemical having a poly-dispersity of 1.15 for both Mw = 4.1 x 106 and7.1 x 106.

This exponent value is very sensitive to polydisper-

555

Fig. 7. - Molecular weight dependence of the relative

viscosity from Toyo Soda samples, at c = 5 x 10 - ’ g/g [41].

sity. In fact if we use the viscosity average molecularweight Mv, we find instead of 3. l, an exponent valueof 3.5. There is no reason to use the Mv mean value.Making the assumption that the mean longest relaxa-tion time of a polydisperse sample is the weight averageof TR [6], since the elastic modulus is independent of themolecular weight, we have :

where wi is the weight fraction of species having amolecular weight M;, the mean molecular weight tobe used is closer to M,, than Mv.

In fact, to our knowledge, most of the authors [33],- except references [21, 34] - measured the viscositymolecular weight dependence using the M, meanvalue. They determine the molecular weight of theirsamples through the Mark Houwink equation.The molecular weight exponent value (3.1) seems

to be closer to reptation prediction (3) than the 3.5Bueche prediction [35], in the case of high molecularweight (M > 106).

4. Conclusion. - These sets of experiments havedemonstrated a certain number of facts concerningthe viscosity behaviour of semi-dilute solutions forwhich the monomer concentration is less than 10 %but still higher than c*.

a) The relative viscosity does not depend on theglass transition temperature of the undiluted polymer.

b) Since the temperature variation of a 9 % PS-

benzene solution, having a ratio of -c I IS - identicalc * 2

to that of the solvent viscosity, we may conclude thatthe local viscosity is that of the solvent. This confirmsthe NMR experimental results [36].

c) The relative viscosity of semi-dilute solutionin good solvent, seems to be, in the case of PIB-toluene,in agreement with de Gennes theory. This fact confirmsthe experimental results [21] obtained a long time ago.

d) The viscosity of the PS-benzene and PIB-cyclo-hexane seems to be influenced by local conformation.

e) The viscosity at the 0 temperature is much morecomplex than in good solvent - scaling laws do notseem to be applicable.

f ) The molecular weight dependence of the viscosity(~ M3-1) is close to reptation model.

It appears that the concentration dependence of therelative viscosity could be interpreted using deGennes’ theory of semi-dilute solutions. The problemposed by these experiments is the decrease of the

viscosity when the temperature increases. The classicaltemperature dependence of the characteristic lengthcould not account for this variation.

Acknowledgments. - The authors gratefully thankJ. P. Cohen Addad for the Precious PIB-sample;P. G. de Gennes, P. Pincus and R. Ball for helpful andstimulating discussions.

Appendix 1

Let us suppose that a chain follows - on m mono-mers a local rigidity :- b = m. a, a being the length of a monomer and b

the persistence length,- on gt monomers a Gaussian conformation

whose end to end distance 0 is :

Using the Flory equation for the expansion factor,or following reference [29], near 0 temperature, thevariation of g, with the reduced temperature r.

is

In part 1, we have considered that the effective

length of a monomer was defined through ç = lg’and that the excluded volume effect is effective at anyscale. In particular

(A 1) and (A. 3) are compatible only if :

Substituting this expression of I (A. 4) in the for-mula (2) giving ç as a function of c and I we obtain :

usingand

556

and

Usually m = U jkT, k is the Boltzmann’s constantand U the bending energy per monomer. The tem-perature dependence of the quantities g/gt and q, are :

Appendix 2

NUMERICAL VALUES CORRESPONDING TO FIGURES 2, 3,6, 7 ; THE RELATIVE VISCOSITY ACCURACY IS 2 %.

The. viscosity of the solvent was calculated usingthe values given in reference [17]. We obtain the

following relations :

Appendix 3

TEMPERATURE DEPENDENCE OF THE RELATIVE VISCOSITY

OF PIB-CYCLOHEXANE SYSTEMS AT DIFFERENT CONCEN-TRATIONS.

References and notes

[1] BERRY, G. C. and Fox, T. G., Adv. Polym. Sci. 5 (1968)261.

[2] DE GENNES, P. G., Scaling concept in polymer physics(Cornell University Press) 1979.

[3] c* ~ N/R3 , N is the number of monomers in a chain ofradius R. The concentration in section 1 is express-ed in volume number density of monomers, insections 2 and 3.3 is expressed in g/g

[4] EDWARDS, S. F., Proc. Phys. Soc. 92 (1967) 9.[5] DE GENNES, P. G., Macromolecules 9 (1976) 587-594.[6] DAOUD, M., DE GENNES, P. G., J. Polym. Sci. Phys. Ed.

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[9] LE GUILLOU, J. C., ZINN-JUSTIN, J., Phys. Rev. B 21(1980) 3976.

557

[10] Here we do not make a distinction between the staticexponent, v, and the effective dynamic exponent,03BDH, because the latter plays a minor role in theconcentration exponent value.

[11] DAOUD, M., JANNINK, G., J. Physique 37 (1976) 973.COTTON, J. P., NIERLICH, M., BOUÉ, F., DAOUD, M.,

FARNOUX, B., JANNINK, G., DUPLESSIX, R., PICOT,C., J. Chem. Phys. 65 (1976) 1101.

[12] ADAM, M., DELSANTI, M., J. Physique 41 (1980) 713.[13] One defines the 03B8 temperature as the temperature at

which the second virial coefficient is zero, for aninfinite dilute solution.

[14] BROCHARD, F., DE GENNES, P. G., Macromolecules 10( 1977) 1157.

[15] ADAM, M., DELSANTI, M., J. Physique Lett. 40 (1979)L-523.

[16] Details on the apparatus will be published :ADAM, M., DELSANTI, M., MEYER, R., PIERANSKI, P.,

J. Phys. Appl.[17] For each solvent we have used the viscosity temperature

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BRANDRUP, J., IMMERGUT, E. H., Polymer Handbook(1975).

[18] ADAM, M., DELSANTI, M., Macromolecules 6 (1977)1229.

[19] PS-benzene and PS-cyclohexane :DECKER, D., Thesis, Strasbourg (1968).PIB-cyclohexane and PIB-03B8 solvent (IAIV) :MATSUMOTO, T., NISHIOKA, N., FUJITA, H., J. Polym.

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molecules 13 (1980) 1280.[23] We are indebted to R. Ball for pointing out this effect.[24] WEILL, G., DES CLOIZEAUX, J., J. Physique 40 (1979) 99.

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[25] SCHMITT, A., J. Physique Lett. 40 (1979) L-317.

[26] DAOUD, M., COTTON, J. P., FARNOUX, B., JANNINK, G.,SARMA, G., BENOIT, H., DUPLESSIX, R., PICOT, C.,DE GENNES, P. G., Macromolecules 8 (1975) 127.

[27] HERVET, H., LÉGER, L., RONDELEZ, F., Phys. Rev. Lett.42 (1979) 1681.

[28] We know from reference [12] that the concentrationexponent value of the cooperative diffusion coeffi-cient is insensible to the exact value of effective

exponents 03BDH and v.[29] ISONO, Y., NAGASAWA, M., Macromolecules 13 (1980)

862.

[30] DE GENNES, P. G., J. Physique Lett. 35 (1974) L-133.[31] ROOTS, J., NYSTRÖM, B., Macromolecules 13 (1980) 1595.[32] VIDAKOVIC, P., ALLAIN, C., RONDELEZ, F., J. Physique

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ALLEN, V. R., Fox, T. G., J. Chem. Phys. 41 (1964) 337..Fox, T. G., FLORY, P. J., J. Am. Chem. Soc. 70 (1948)

2384.

DREVAL, V. E., MALKIN, A. YA, BOTVINNIK, G. O.,J. Polym. Sci. 11 (1973) 1055.

ZAPAS, L. J., PADDEN, F. J., DEWITT, T. W., Specialreport of Federal Facilities Corporation (1954).

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[34] RUDD, J. F., J. Polym. Sci. 44 (1960) 459.SHAW, M. T., Polym. Eng. Sci. 17 (1977) 266.

[35] BUECHE, F., J. Chem. Phys. 25 (1956) 399.[36] COHEN ADDAD, J. P., GUILLERMO, A., MESSA, J. P.,

Polymer 20 (1979) 536.[37] STRAZIELLE, C., BENOÎT, H., Macromolecules 8 (1975)

203.

[38] EICHINGER, B. E., FLORY, P. J., Trans-Faraday Soc. 64(1968) 2053.

[39] KRIGBAUM, W. R., FLORY, P. J., J. Amer. Chem. Soc.75 (1953) 1775.

[40] FERRY, J. D., Viscoelastic properties of polymer (J.Wiley, N.Y.) 1970, p. 316.

[41] For numerical values corresponding to figures 2-3-6-7,see appendix 2.