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Viscosity and longest relaxation time of semi-dilutepolymer solutions. I. Good solvent

M. Adam, M. Delsanti

To cite this version:M. Adam, M. Delsanti. Viscosity and longest relaxation time of semi-dilute polymer solutions. I. Goodsolvent. Journal de Physique, 1983, 44 (10), pp.1185-1193. �10.1051/jphys:0198300440100118500�.�jpa-00209702�

1185

Viscosity and longest relaxation time of semi-dilute polymer solutions.I. Good solvent

M. Adam and M. Delsanti

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

(Reçu le 10 mai 1983, accepté le 27 juin 1983)

Resume. 2014 Des mesures de viscosité à gradient de cisaillement nul ~, et de temps de relaxation le plus long TRsont effectuées dans des solutions semi-diluées de polystyrène-benzène. Lorsque la concentration c est inférieureà 10 %, dans une grande gamme de concentration réduite (4 c/c* 70), c* étant la concentration de recou-vrement, nous avons trouvé que :

2014 la viscosité relative ~r ainsi que le temps caractéristique le plus long TR divisé par le temps caractéristique dupremier mode T1 d’une chaîne unique sont des fonctions de la concentration réduite c/c* seulement,

2014 le module élastique de cisaillement G est indépendant de la masse moléculaire.Ces résultats sont en accord avec les prévisions théoriques.Nous trouvons que les variations de ~r TR avec la concentration réduite c/c*, et avec la masse moléculaire, sont :

Toutefois, nous trouvons que l’exposant Xc est une fonction croissante de c/c* et que XM croît avec la concen-tration. Ces résultats ne peuvent être expliqués par un modèle de reptation classique.

Abstract. 2014 The zero shear viscosity and longest relaxation time are measured for semi-dilute polystyrene benzenesolutions. For monomer concentrations c smaller than 10% and over a large range of the reduced concentrationc/c* where c* is the overlap concentration (4 c/c* 70), we find that :

2014 both the relative viscosity ~r and the longest relaxation time TR divided by the first mode characteristic timeT1 of a single chain are function of the reduced concentration c/c* only;

2014 the shear elastic modulus is independent of the molecular weightThese results are in agreement with theoretical predictions. We find that ~ and TR depend on the reduced con-

centration and the molecular weight as follows :

However, we find that Xc is an increasing function of c/c* and that XM increases with c. This cannot be explainedusing the simple reptation model.

J. Physique 44 (1983) 1185-1193 OCTOBRE 1983,

Classification

Physics Abstracts46.30J - 52.40 - 61.40K - 82.90

1. Introduction.

De Gennes’ theory of the viscosity of a semi-dilutepolymer solution has already been reviewed [1, 2];here we recall the main results.The asymptotic semi-dilute regime is defined by a

monomer concentration c which is smaller than an

upper concentration c, and greater than the overlapconcentration c* :

In dilute solution, a chain with a molecular weight M,has a radius of gyration, R related to M as follows :

with v = 0.588.The zero shear viscosity of the solution ’1 is pro-

portional to the solvent viscosity ’10 and is a functionof the reduced variable c/c* only. If clc* > 1, therelative viscosity tl, = ’11 ’10 obeys a power law :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198300440100118500

1186

The molecular weight dependence of the viscositypredicted by reptation theory [3] (XM = 3), imposesthe Xc exponent value :

Thus :

The longest viscoelastic relaxation time of the

polymeric system TR, called the reptation time, is thetime needed for a chain to reptate the length of afictitious tube. It represents the hindrance to thechain motion due to the presence of entanglements.This time is proportional to the first Zimm mode T,of the single chain, and is a function of c/c*, only.

If c » c*, .

with

A relation exists between the viscosity ?I and thelongest relaxation time TR :

where G is the shear elastic modulus.

2. Experimental conditions.

2.1 SAMPLE PREPARATION. - The solvent used isbenzene of analytical grade (RP) and the polymer ispolystyrene.The monomer concentration [4] of a semi dilute

solution lies between the overlap concentration c*and an upper concentration S When the monomerconcentration exceeds i (e - 10 % for polystyrenebenzene) the solutions no longer follow the lawsobtained for the semi-dilute case. Actually, the polymermutual diffusion coefficient is an increasing functionof concentration [5] in the polystyrene ethylbenzenesystem up to 20 %, where it becomes stationary beforedecreasing. The correlation function of the lightscattered by a polystyrene benzene solution has anexponential line shape only if c 10 %. In additionthe benzene mutual diffusion coefficient [6] has anactivation energy independent of concentration, onlyif c 10 %.Our usual definition [7] of the overlap concentration

c*, for polystyrene benzene solutions, is :

our definition of c* corresponds to the concentrationat which the osmotic pressure [8] no longer depends

on the molecular weight, a behaviour which is cha-racteristic of semi-dilute solutions.

In order to fulfill the condition c* c 10 %over a large concentration range we use high mole-cular weight polystyrene whose characteristics aregiven in table I. Some measurements were also done onlow molecular weight polystyrene, at high concen-tration (see Table I).

Samples are prepared in the measuring cell a longtime in advance (at least two months). No mechanicalforce is exerted on the polymer in order to facilitatethe dissolution; we check that there is no concen-tration gradient by measuring the viscosity at variouspoints of the cell.

Table I. - Molecular weight, polydispersity index andoverlap concentration c* of the samples studied. Poly-styrenefurnish by : (-) Toyo Soda inc. (Japan), (*) CRMStrasbourg (France).

2. 2 APPARATUS. - In order to measure the zero shear

viscosity and the longest relaxation time we use amagnetorheometer described in reference 9. Herewe only recall the main features. A magnetic sphereimmersed in the sample experiences a magnetic forceprovided by a current passing through a coil. Thecurrent intensity I is monitored by a feedback ampli-fier, so that it maintains the sphere at a fixed position.When the sample is displaced by a step motor at aspeed v, the measurement of the current intensity Ipassing through the coil allows us to determine theviscous force :

where Io is the current intensity needed to counter-balance the gravitational force and r is the radius ofthe sphere.We measure a zero shear viscosity because the

ratio - does not depend on the shear rate vlr.v

p

All our experiments are performed at a reduced shear

rate v x TR ’) smaller than 10-2.

We r t¿e absolute value of the viscosity of ourWe obtain t e absolute value of the viscosity of oursamples by a calibration of our apparatus with stan-dard silicon oil (’1 = 40.47 poises at 35 OC).

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A range of viscosities, between 0.1 and 101 poises,are measurable within a precision of 2 %. Therelative viscosity q, is calculated using for the benzeneviscosity 110 :

obtained from tabulated values [10].This apparatus allows us also to measure the

longest relaxation time of the polymeric system. Afterthe motor is stopped, the variation of the currentintensity I of the coil is analysed as a function of time.In order to minimize the noise, the signal is accumu-lated on a multichannel analyser, triggered at themoment that the motor is stopped (Fig.1).The profile of the curve is fitted to an exponential

line shape :

and the parameters A, TR and B are determined by aleast mean square program. We check that the cha-racteristic time TR is, to within experimental accuracy(10 %), independent of :- the characteristic coefficients of the electronics

which monitor the current intensity of the coil,- the speed v of the sample before the motor is

stopped,- the time per channel of the multichannel.

The range of relaxation times which are measured

using the magnetorheometer is between 120 s and50 ms. The lower limit is imposed by the presentcharacteristic coefficient of the electronics and by theamplitude of the signal which, being proportional tothe macroscopic viscosity, becomes too small to bedetected when the viscosity is smaller than 2 poises.

Fig. 1. - Variation of the current intensity passing throughthe coil in arbitrary units as a function of time. The full lineis the best fit corresponding to a characteristic time of0.184 s (the sample is : Mw = 6.77 x 106, c = 4.2 % at35 oC).

The experimental treatment, using a single expo-nential decay for the current intensity, implies thatthe longest relaxation time is much larger than all theothers characteristic times of the polymeric system.

Foliowing the Doi-Edwards and de Gennes theories[11], the relaxation times for a reptating chain are :

where p is an odd number of modes whose amplitudesvary as l/p2. Therefore, experimentally the system issensitive to the first mode only since the second modehas an amplitude and a characteristic time which is10 times smaller than the first one.

In the semi-dilute polymeric system there is also aspectrum of relaxation times with first mode [1]Tç ç3, ç being the screening length of hydrody-namic and thermodynamic interactions. One can

easily show that TR/Tç (clc*)’-". Thus under ourexperimental conditions (c > 4 c*) the reptation timeTR is much larger than Tj.The experimental results obtained, concerning the

shear elastic modulus (see section 3 .1. 3) allow us to beconfident that we are measuring the longest relaxationtime with :

The sample is thermalized by water circulation;the homogeneity and the stability of the temperatureis better than 0.1 OC.With this magnetorheometer we have therefore

measured simultaneously the zero shear viscositv

(0.1 poise ri 105 poises) and the longest relaxationtime (5 x 10- 2 s TR 100 s).

3. Experimental results.

Two sets of experiment were carried out in order totest :

- the concentration dependence of ", T R and G.Using several molecular weights, the monomer con-centration was increased from c* to 10 %. For thetwo lowest molecular weights, the experiments wereperformed up to 20 %.- the molecular weight dependence of q and TR.

At 3 given concentrations (2, 5 and 8 %) the molecularweight was increased from 1.26 x 106 to 20.6 x 106.

In sections 3.1 and 3.2 we give the experimentalresults obtained in a good solvent (benzene) for ",TR, and G. They are discussed in section 3 . 3. The rawresults are given in the appendix.

3.1 CONCENTRATION DEPENDENCE.3.1.1 Relative viscosity. - Whatever the molecularweight, in the range 1.7 x 101 and 20.6 x 106, therelative viscosity depends only on a reduced variable :the reduced concentration c/c*. In figure 2, one can see,that the relative viscosity, for a wide range of c/c*

1188

Fig. 2. - Relative viscosity as a function of the reducedconcentration (log-log scale). The following symbols areused for the different molecular weights : Mw = 20.6 x 106 8,6.77 x 101 0, 3.84 x 106 EL 2.89 x 106 *, 1.26 x 106 +,4.22 x 10’ x, 1.71 x 10’ V.

values (2 to 70), is independent of the molecularweight and is a function of c/c*, only.The experimental data obtained at c/c* > 4 lead to :

Using equation 8, c* _ M-O.785, we obtain :

This result (Fig. 2), although spectacular, couldpossibly be only qualitative because discrepanciescan be hidden by the large variations of the observedviscosity. To get rid of this difficulty we use anotherrepresentation : we consider the ratio q/%, where q,is the measured quantity and % = 0.35(c/c*)4.07.From figure 3, where flr/ is plotted as a function ofc/c*, we observe that for the highest molecular weights(M > 106), the points obtained with different mole-cular weights do not deviate systematically from eachother.

Fig. 3. - Variation of the measured quantity tl, divided by% calculated using equation 9. Symbols are the same as infigure 2.

Thus c/c* is the correct reduced variable for il, and :

where the effective exponent X(c/c*), defined by :

reaches a constant value only if c > 10 c*.In order to test the variation of the effective expo-

nent with molecular weight we determine X(c/c*)discretely or, when the number of experimental pointsallows, continuously. Figure 4 gives the variationof X(c/c*) calculated for each molecular weight as afunction of c/c*. The full line corresponds to theresults obtained with the highest molecular weight(M > 1.26 x 106) at low concentration (c 10 %).We observe that the effective exponent X(c/c*)increases from 2 to 4.5 over the whole range of con-centration studied and that the 4.07 concentration

exponent value (see Eq. 9) found previously corres-ponds to a mean value of X(clc*). A power law fit to theresults obtained at c/c* > 10, leads to a stable exponentvalue equal to 4.46 ± 0.05 in agreement with previousresults [12]. In figure 4, the points corresponding to theresults obtained with the two lowest molecular weightsat a concentration lying between 8 and 20 % aresystematically above the full line.We have also considered the variation of X(clc*),

obtained at given values of c/c*, as a function ofconcentration. The variation of Xe with concentrationgives rise to a family of curve each one correspondingto a value of c/c*. Each curve is normalized by itsvalue Xe*e-+O obtained at the lowest concentration,the family of curve is then reduced to one (Fig. 5).At a concentration of about 10 %, the normalizedquantity X, .IX,*,,-o no longer has a constant value

Fig. 4. - Variation of concentration exponent X c as afunction of c/c*, the full line represents Xc obtained withhighest molecular weights, the arrow indicates the theoreticalvalue. Symbols are the same as in figure 2.

1189

Fig. 5. - Variation as a function of concentration (in g/g)of the effective exponent Xc divided by its value X,,.,c-Oobtained from the semi-dilute regime and extrapolated tozero concentration. Symbols represent different values ofc/c* : c/c* = 18., 7.38 0, 5 x, 3.5 EL 2.5 p.

(Fig. 5). This concentration corresponds to the upperconcentration c.These observations will be discussed in sec-

tion 3. 1. 3.

3.1.2 Longest relaxation time. - Due to the lowerlimit of the relaxation time TR measurable with thisapparatus (50 ms), the measurements can only bedone on samples having a reduced concentrationgreater than 7, thus on a smaller range of concentrationthan in the preceding section. In all cases, the currentintensity in the magnetic coil relaxes to equilibriumexponentially (see Fig, 1).As an example, in figure 6, we give, the variation of

TR as a function of concentration, obtained with themolecular weight 20.6 x 106. Typically a samplewhose concentration is 2.66 % has a longest relaxationtime of 3.34 seconds.

Following de Gennes’ predictions (see introduc-tion) : .- the ratio, TRIT,, of the longest relaxation time

Fig. 6. - Variation of the longest relaxation time as a func-tion of concentration in g/g (log-log scale) obtained usingMw = 20.6 x 106. The corresponding exponent value is :2.18 + 0.1.

to the time of the first Zimm mode of a single chain T1,is a function of the reduced concentration, c/c*, only(see Eq. 6).- In the asymptotic semi-dilute regime (c >> c*),

this function is a power law, with exponent Y c (obtainedwith v = 0.588) equal to 1.6.

We calculate T 1 using the relation :

where kB, T and A are the Boltzmann constant, theabsolute temperature and Avogadro’s number, respec-tively ; a, a numerical constant which is taken to beequal to 0.42 as in the case of a gaussian chain withhydrodynamic interactions [13]. [tl], is the intrinsic

viscosity, which, in the case of polystyrene benzenesolution [14], is :

The ratio TRI T, is plotted as a function of thereduced concentration c/c* (Fig. 7), yielding a singlecurve independent of the molecular weight and

dependent on c/c*, only. The fit to the experimentaldata gives :

The important experimental result obtained fromfigure 7 is that the reduced concentration c/c* is areduced variable for the ratio TRIT,, in agreementwith de Gennes’ theory (Eq. 6). However the experi-mental value of Yc = 2.05 is much higher than thetheoretical value (1.6).

This point will be discussed further in section 3 .1. 3.

Fig. 7. - Log-log plot of the reduced relaxation time TRI T 1,as a function of c/c*. The slope corresponds to the exponenty c = 2.05 ± 0.1 (see Eq. 11). Symbols are the same as infigure 2.

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3.1.3 Elastic shear modulus. - From our experi-mental determinations of the viscosity, the longestrelaxation time, and the relation 7 : G = j7/ TRI wededuce the shear elastic modulus.Our determination is in good agreement with

experimental results obtained with a relaxometerof the cone and plate type [ 15] on a polystyrene aroclorsolution (Mw = 8.42 x 106).We note that the shear modulus is much smaller

that the osmotic bulk modulus defined by :

The osmotic bulk modulus and osmotic pressure 7rhave the same concentration dependence.From osmotic pressure measurements [8], on poly-

a-methylstyrene dissolved in toluene it is found that7t - C2.32 and that at a concentration of 7.8 g/cm3corresponding to a reduced concentration c/c* of 18.2 :

At the same concentration the shear modulus valueis 2 x 103 dynes/cm2. Thus these two moduli differby a factor of 140.From figure 8, where the shear modulus G is plotted

as a function of concentration [16] we observe that :- G is independent of the molecular weight- G is only concentration dependentWe found that :

with

Thus the shear elastic modulus and the osmotic bulkmodulus are proportional. They vary with concen-tration with the same exponent value (2.36 and 2.32,

Fig. 8. - Log-log scale of the shear elastic modulus as afunction of concentration expressed in g/cm3 [16]. The slopecorresponds to the exponent Zc = 2.36 (see Eq. 13). Symbolsare the same as in figure 2, the ® is the value of the shearmodulus extracted from reference 15.

respectively), they differ only by a large numericalfactor.

It has been shown [1] that in a semi-dilute solutionthe osmotic pressure is a function of the correlation

length of the density correlation function :

then n - K = G - C3v/(3v-l).Using the experimental determination of the expo-

nent Z, and equation 14 we find : v = 0.58; in goodagreement with v values determined previously [17].We note that the relation 7 :

implies that the concentration exponents of il, Gand TR are related by : X, = Y,, + Z,. This is onlyfound to be true if each exponent is determined overthe same range of concentration.

Actually for c/c* > 7 :

3.2 MOLECULAR WEIGHT DEPENDENCE OF THE VISCO-SITY AND LONGEST RELAXATION TIME. - The mainresult of the classical reptation model [1, 18] is that,when M > Me in a polymer melt, or when c > c*in a semi-dilute solution, the molecular weight expo-nent value (see Eq. 3) (il - TR MXM) is equal to 3,whatever the concentration and the quality of thesolvent

This theoretical result is not obeyed in a polymermelt where it is well lnown that the exponent valueis around 3.4 [19, 20].

In order to measure the XM exponent in semi-dilutesolutions we prepared samples with given concen-trations (2, 5 and 8 %) using five different molecularweights (M > 1.26 x 106). At a given concentrationthe variation of the viscosity and of the longestrelaxation time as a function of molecular weightallows us to determine XM.

Figure 9 shows the experimental results obtainedfrom samples with a monomer concentration of 8 % ;one observes that the exponent XM is well defined andits values from 17 (curve a) and TR (curve b) measure-ments are XM = 3.38 ± 0.04 and 3.32 ± 0.07, respec-tively.

Moreover, whatever the given concentration, largerthan c*, and whatever the temperature, both XMdeterminations (via il, or TR) lead to the same expo-nent value. Obviously, this corresponds to the resultalready mentioned above : namely that the shearelastic modulus (G = 17/T R) is independent of themolecular weightAt a given temperature the exponent XM is concen-

tration dependent This can be observed in figure 10where the XM values are plotted as a function of

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Fig. 9. - Variation of the relative viscosity (curve a) and ofthe longest relaxation time (curve b) as a function of mole-cular weights (log-log scale). The concentration of samplesis 8 %. The slope corresponds to the XM exponent (see Eq. 3).

Fig. 10. - Variation of the molecular weight exponentvalue XM as a function of concentration, the experimentis carried out at a temperature of e 35 °C and x 60°C.

concentration, XM is bigger than 3 and increases withconcentration although all the measurements are doneat c > 4 c*.

In fact this dependence of the XM exponent value onconcentration is already contained in the precedingobservation (Eq. 9) :

where the effective exponent X(clc*) increases withthe reduced concentration.The relative viscosity decreases with increasing

temperature and at a given concentration (8 %) theXM value obtained from measurements done at atemperature of 60 OC is slightly smaller (3.28) thanthe XM value obtained at room temperature (35 OC).

3.3 DiscussioN. -We now discuss experimentalresults which cannot be accounted for by using theclassical reptation and semi-dilute polymer solutionmodel. The concentration exponent X (c/c*) is a

function of the reduced concentration c/c* if c 10 %,or equivalently the molecular weight exponent is alsoa function of the absolute concentration if c e - 10 %.

Since the viscosity measurements are performedover the widest range of c/c* (2 c/c* 70) we willdiscuss essentially these results.

Considering the case of a polymer solution whoseconcentration is smaller than 10 %, below a value ofc/c* of the order of 5 the exponent Xe is smaller thanthe theoretical value (3.92). This can be due to a largecross-over domain between dilute and semi-dilutesolutions for viscous properties.Above this c/c* (~5) value, the exponent Xr

increases to a value 4.5. Let us examine the two

arguments which can explain the concentration depen-dence of Xc.The first one it the well known Weill-des Cloizeaux

[21] calculation taking into account the local gaussianconformation of the chain. This assumption, coupledwith a local rigidity was analysed in detail in reference 2for the case of a semi-dilute solution.

The second one is the influence of the finite monomerconcentration. In de Gennes’ theory it is supposedthat the local friction is only due to monomer solventfriction. At high concentrations one must introduceanother local friction : the monomer monomer frictionwhich increases the concentration dependence of theviscosity.

It seems to us that both arguments (local gaussianconformation and monomer monomer friction) willlead to an increase of the concentration exponent withthe absolute value of concentration. Therefore theycannot explain the increase of the concentration

exponent X(clc*) with the reduced concentration.However this can explain the high value of the expo-nent found at high monomer concentration

(c > 1 - 10 %). Fortuitously, X, is then nearer tothe theoretical value than are the X e values found atthe same c/c* value but at low concentrations (c j).This could explain why by self-diffusion coefficientmeasurements one determines [22, 23] concentrationexponents identical to theoretical values although theexperiments are performed at high concentrations(2 % c 40 %) but at low value of c/c* (clc* Z 7).Another way of discussing the increase of X(c/c*)

with the reduced concentration is to analyse thevariation with concentration of the molecular weightexponent XM.We determine the XM values (see Eq. 3) using the

same five different molecular weights, modifying theconcentration only. Therefore our experimental resultsrule out our assumption that the high value of XMfound experimentally in a polymer melt, was an effectof polydispersity [2].Two theoretical interpretations have been given in

order to explain a value of the exponent XM (see Eq. 3)bigger than 3.The first interpretation due to Doi [24] accounts for

the fluctuations of the chain inside the tube. He finds

1192

that the viscosity of a polymer melt increases fasterat M Jit Me than at M >> Me (Me being the molecularweight between two consecutive entanglements). Wehave used the expression proposed by Doi

where MIM,, is replaced by N/g - (cjc*)1.25 whereg(- Cç3) is the number of monomers between twoconsecutive entanglements, and in the numericalcalculation c/c* and M values are taken identical tothose used in the experiment.We found that : at c = 2 %, X m = 3.64,

and at c = 8 %, X’ = 3.26.Thus XM decreases as c increases.On the other hand, using the Doi expressions for r¡

and TR one finds a shear elastic modulus which ismolecular weight dependent.

Thus, the results of this calculation are not verifiedby experiments done in semi-dilute solutions.The second interpretation, due to Wendel and

Noolandi [24], allows for hindered defects diffusingalong the chain, the usual diffusion law ( r2(t) > "-I t

is then replaced by : ( r2(t) > "-I tl-lwhereo , y 1.The authors find a reptation time : T R "-I N(N 2)2/(l - y)which corresponds to the classical reptation time wheny = 0 and to an infinite reptation time when y = 1(the chain is trapped by defects).

If one supposes that this exponent y increases withthe number of entanglements (~ c/c*) and decreaseswith temperature, this hindered motion can explainour results concerning the increase of the concentrationexponent as a function of the reduced concentrationand the increase of the molecular weight exponentwith concentration. However one must understandwhich physical mechanism is responsible for thishindered motion leading to a slowing down highenough to be measurable.

4. Conclusion.

The main results of de Gennes’ theory - on theviscosity and reptation time of semi-dilute polymersolutions are well verified by the experiments :- The relative viscosity is a function of the reduced

concentration c/c*, only. The concentration exponent(4.07) and the molecular weight exponent (3.2) deter-mined over the whole range of concentration

(4 c/c* 70) is not far from the theoretical values.- The reptation time divided by the first Zimm

mode characteristic time of a single chain is also afunction of the relative concentration only. Howeverthe concentration exponent value (2.05), determined at7 c jc* 70 is higher than the theoretical value (1.6).- The shear elastic modulus is independent of the

molecular weight and dependent on concentrationonly. The concentration exponent value (2.36) isidentical to the value determined by osmotic pressure

measurements, in agreement with theoretical pre-dictions.

However if the experimental results are carefullyanalysed we find that they cannot be wholly describedusing de Gennes’ and the classical reptation model.

Actually we find that :- at high concentration (c > 6 - 10 %) monomer

monomer friction plays a role ;- at lower concentration (c c ) we observe an

increase of the concentration exponent value with thereduced concentration c/c* and a molecular weightexponent, bigger than 3, which increases with concen-tration.

In the reptation model, applied to semi-dilutesolution, it is supposed that the diffusion coefficientof the chain in the tube is anisotropic - it is nullwhen one considers the motion perpendicular to thetube, it is finite and easily calculated when one consi-ders the motion parallel to the tube. The diffusioncoefficient D is inversely proportional to the numberof entanglements.Our experimental results could be explained by a

decrease of D jj due to an additional dissipative effecttaking place at the entanglements [26].

Acknowledgments.

, The authors gratefully thank J. M.. Bernal and M.Labouise for technical assistance; P. G. de Gennes,M. Daoud and K. Osaki for helpful discussions.

Appendix.Raw results of the zero shear viscosity 17 and the longestrelaxation time TR of polystyrene benzene solutionsat a temperature of 35 OC.

Polystyrene Mw = 3.84 x 106

1193

Polystyrene

Polystyrene Mw = 20.6 x 106

Polystyrene Mw = 6.77 x 106

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[17] See for instance : LE GUILLOU, J. C., ZINN-JUSTIN, J.,Phys. Rev. B. 21 (1980) 3976.

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