Intrinsic viscosity of flexible macromolecules : transition from theta to collapsed state

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Intrinsic viscosity of flexible macromolecules : transition from theta to collapsed state

R. Perzynski, M. Delsanti, M. Adam

To cite this version:

R. Perzynski, M. Delsanti, M. Adam. Intrinsic viscosity of flexible macromolecules : tran- sition from theta to collapsed state. Journal de Physique, 1984, 45 (11), pp.1765-1772.

�10.1051/jphys:0198400450110176500�. �jpa-00209918�

Intrinsic viscosity ^of flexible macromolecules : transition from theta to collapsed ^state

R. Perzynski (*), ^M. Delsanti and M. Adam Service de Physique ^{du Solide et} de Résonance Magnétique, CEN-Saclay, 91191 Gif-sur-Yvette Cedex, ^France

Résumé.

Des mesures de viscosité intrinsèque [~] ^sont effectuées sur des solutions diluées de polystyrène ^{dans le} cyclohexane ^en fonction de la température (T ~ 0) ^{et de la masse} moléculaire (8,2 ^x 104 ~ Mw ~ 2,06 ^x 107).

Le facteur d’expansion (03B13~ ^{= [~]/[ ~03B8])} de la viscosité intrinsèque ^{obéit à une loi} d’échelle dont la variable réduite est

(T 2014 03B8/T) · ~Mw ^Lorsque (T 2014 03B8/T) · ~Mw ~ 2014 40, la viscosité intrinsèque ^d’une chaîne effondrée est indépen-

dante de la masse moléculaire. L’indice 03BD reliant la taille à la masse moléculaire est trouvé égal ^à 1/3.

Abstract.

Intrinsic viscosity measurements have been performed ^on polystyrene-cyclohexane system ^{as a func-} tion of temperature T (T ~ 03B8 and as ^{a function} of molecular weight (8.2 ^x 104 ~ Mw ~ ^{2.06 x} 107) .The expan-

sion factor 03B13~ ⁼ [~]/[~03B8] of the intrinsic viscosity [~] scales with the reduced variable (T 2014 03B8/T) · ~Mw. ^For

(T201403B8/T) ” ~Mw ~ 2014 40, the intrinsic viscosity ^{of a} collapsed macromolecule is found to be independent ^of

molecular weight ^{A value 03BD =} 1/3 is therefore given ^{to the index} relating ^{size to molecular} weight.

(Recu le 24 avril 1984, accepté ^{le 4} juillet 1984)

Classification

Physics ^Abstracts

47.50

61.40K - 64.40K

Introduction.

Many experimental [1-10] works have been devoted to the collapse ^{of a flexible} polymer ^chain using ^a

great number of different techniques, ^each experiment being performed ^{on a} limited molecular weight

range. They ^{lead to} puzzling ^results [11]. Although

intrinsic viscosity ^is very sensitive to chain contraction,

as it is proportional ^{to a volume, few} measurements [7, 12-14, 9] have been carried out systematically ^to study ^the collapse. ^{We have} performed ^{such measure-}

ments over a large range of molecular weights (M, ^{= 8.2 x 104} up ^to 2.06 ^x 10’). ^{We use the} polystyrene-cyclohexane system below the 0 tempe- rature, ^{in the dilute} regime, ^with sample ^concen-

trations as low as possible ^{in order to observe} single

chain behaviour over a temperature range ^as wide as

possible.

In the first part ^{of this} paper ^we recall briefly ^the

theoretical predictions of the intrinsic viscosity. ^{In the}

second part ^we describe the experimental conditions.

In parts three and four we present and discuss our

results, respectively.

1. Theoretical background

From the specific viscosity _r¡sp ^{of a} macromolecular

solution, ^{a molecular} quantity ^{can be derived :} [ r], ^{the intrinsic} viscosity ^{of one} macromolecule.

tisp ^{is defined as :}

with n being ^the viscosity ^{of the} solution, flo, the

viscosity ^{of the solvent, C, the} monomer concen-

tration of the solution, M, the molecular weight ^of

the macromolecule, ^and T, ^the temperature. ^In dilute solution flsp is written as a second order _expan- sion in concentration [15] :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198400450110176500

1766

Usually [15] K17 =KH’ 1171, ^with KH being ^the Huggins

interactions coefficient. From homogeneity argu- ments [16-18], [’1] ^{can be written as :}

where R is the end to end distance of the macro-

molecule which scales with M as M’, ^then [11] _ ^{M3v-l. At the 0} temperature, where thermo-

dynamic interactions vanish, the macromolecule obeys

Gaussian statistics, ^with v = 1/2 ^and [ n]o - Ro - MO - 5.

For the polystyrene-cyclohexane system ^{when the} temperature ^is decreased below 0, thermodynamic

interactions ^are attractive and intrinsic viscosity

decreases because macromolecules contract. This contraction can be described using ^{the modified} Flory equation [18, 19] :

where a is the end-to-end expansion ^factor

c is a constant of order unity, ^{z is the} excluded volume parameter, y ^{is a} coefficient proportional ^{to the}

ternary ^cluster integral frequently ^{related to steric}

hindrance [18-20], ^{which is constant and} independent

of T. This mean field description, ^{as well as} scaling approach, ^{lead to one} universal reduced variable which is [19] :

where v is the binary ^cluster integral, N, ^{the number} of statistical units of length ^I. Usually ^the following assumptions ^{are made :}

-

N is proportional ^{to the molecular} weight M,

-

v is proportional ^to the relative temperature

z - (T - 0)/r,

-

I is a constant independent ^{of N and T.}

Then the reduced variable of the end-to-end

expansion factor is written as :

The intrinsic viscosity expansion factor, defined as

cx I 3 - I’ll. ^{, is also an} universal function of the same

’ hL

reduced variable. OC3 ^{a3 is expected to have the}

following ^behaviour.

1) ^{In the} vicinity ^{of the 0} temperature, the so-called 0 domain, a3 ^{can be developed as a} linear function of

1 jl Experimentally ^it is found that for the hydro- dynamic ^radius [9] ^{the 0} domain extends down to

T, ,IM ^10.

2) ^{In the} collapsed regime ^which corresponds ^to

high negative ^{values of i} JH and ^{a 1 in formula (4),} a£ ^{can be expressed as a power law} in T/Af :

and

In the collapsed regime the volume of the macro-

molecule is proportional ^to M, the intrinsic viscosity

is molecular weight independent and decreases as

the temperature decreases :

2. Experimental conditions and procedures.

The viscosity il ^was measured with a capillary ^{or a} microcapillary viscometer (Viscoboy ^from Lauda)

immersed in a regulated ^{water bath} (see ^the table).

The viscosity ^was deduced from the flow time t in the capillary ^{which was measured} by ^{means of two} photodiodes; ^the reproductibility ^on the flow time

was bt 10-". The specific viscosity ^of the solution

t p y

is 1Jsp ⁼ (t

to)lto [21] with t and to being equal ^to

the flow time of the solution and of the _pure solvent, respectively. ^{We limited our} study ^to temperatures tbwer than 36 °C, slightly above the 0 temperature of the polystyrene cyclohexane system (35 °C). ^At

low temperatures ^the experimental range is limited from below by ^the demixing temperature of the solu-

tion, previously determined by light scattering expe- riments. The temperature ^was measured close to the

capillary by ^a thermometer with an accuracy of 0.05 °C.

We use polystyrene (PS) ^{of the narrowest} poly- dispersity ^available (see ^the table) ^and cyclohexane

from Prolabo without _any purification. ^{We did}

not filter the solutions because, within the experi-

mental _accuracy, the measured specific viscosity ^{of a}

solution (M, ^{= 1.71 x} 105, ^{C = 10-3} g/cm3) ^was

the same with or without a filtration on a 0.22 J.1m

Millipore ^filter.

Measurements were performed ^with sample ^concen-

trations as low as possible. ^{In this} way :

-

we have enlarged ^the temperature range because the demixing temperature is concentration dependent,

-

we have minimized the influence of the inter- action term on 1Jsp (see Eq. (1)).

But as the concentration is lowered 1Jsp ^decreases and the determination of [1J] ^{becomes more and}

more inaccurate. In the case of the highest ^molecular weights, for which measurements with samples ^of

very low concentrations are necessary, ^we used a

differential method. We measured simultaneously

both t and to, ^{at the same} temperature, ^{with two} viscometers immersed in the same bath and previously

calibrated with _pure solvent.

Measurements were performed ^on solutions of con-

Table.

Characteristics of samples ^and capillaries.

MW : weight average molecular weight.

Mw/Mn : polydispersity ^index.

Polystyrene ^furnished by ^{1 : CRM} Strasbourg (France)

2 : Toyo ^{Soda Inc.} (Japan).

f(l) ^{reduced shear rate at 0} temperature ^of capillary ¹ (microviscometer I).

T(2) :

^{capillary 2} (viscometer OB).

f(3)

capillary ³ (viscometer OC).

The viscometers are KPG Ubbelohde viscometers with suspended level bulb from Schott Geriite (West Germany).

The molecular weight symbols ^are those used on figures 4, 5, ^6.

n.m. indicates that no measurement has been performed ^{in the} capillary.

centration C such that 3 x 10-’CIC*3 ^{x 10-2.}

The overlap concentration C * at the 0 temperature is taken to be equal ^to 40/jk ^{(g/cm3) which corres-}

ponds ^{to :}

-

the usual order of magnitude 31[q]o [22, 23],

-

the limit between dilute and semi-dilute beha- viour of the osmotic bulk modulus from light ^scatter- ing measurements [24].

To check experimentally ^the absence of shear rate effect we used different capillaries ^{with same}

solutions. The usual parameter is the reduced shear rate f ⁼ tlo[ q] Mw FIRT ^{where r is the shear rate}

calculated from the viscometer characteristics (see

the Table), R ^{is the} gas ^constant and ilo[il] MW/RT

is proportional ^{to the} relaxation time of the first mode of a single macromolecule. Reduced shear rates at the 0 temperature ^are listed in the table.

All the measurements, except ^{with the 2.06 x} 10’

molecular weight, ^were performed ^with T(9) ^0.3.

No shear rate dependence ^of _17sp was observed at the 0 temperature, ⁱⁿ agreement with reference [25]. ^At

lower temperatures ^some difficulties were encountered,

which we shall discuss in the next section.

Since the capillaries ^{were not} sealed, ^{there was}

a slight evaporation of the solvent when the experi-

ment took a few days. ^{With a} cathetometer we mea-

sured the variation of volume of the solution and then

we controlled the variation of concentration dC.

We made a correction of the specific viscosity

d q,p ^dC

---.:.£ ^{= dC C with DCIC always} smaller than 5 %. ^To

«p

check the validity ^{of the} correction the solution were

always divided in two, and ^we performed :

-

an experiment beginning from 0 and decreas-

ing T,

-

another one beginning ^{with low} temperature and increasing ^T.

After evaporation corrections we found the same

results in both cases.

Measurements, performed with concentrations C such that C ( c > 10-3, ^{lead to an intrinsic} viscosity

accuracy of ⁵ %, ^{the main} error comes from the deter- mination of the concentration. By ^a normalization at the 0 temperature, an ^{is obtained with more preci-}

sion (2 %) :

On the other hand with concentrations such that

C/C* ^{10-3 the main} error comes from the flow time determination and the intrinsic viscosity ^accu-

racy is such that :

1768

3. Experimental ^results.

We take 35 °C as the 0 temperature : ^{it is the} tempera-

ture at which the second virial coefficient of the osmotic pressure vanishes [26]. ^{For the whole} range of mole- cular weight ^and concentration and for the three

capillaries, ^{the theta} specific viscosity ^is always pro-

portional ^{to the} concentration : K,,(O) ^{(see form} (2))

is not detectable and [ tl]o ⁼ ’1sp( C, ^0)/C.

For 2.4 x 101 M, ^{6.77 x 106 we find}

in agreement with other measurements [7, 23]. ^One

must note that a fit to a power low leads to (see Fig. 1) :

This result confirms that :

-

the evaporation corrections are valid,

-

there is no shear rate effect in these measure- ments at the 0 temperature.

Fig. ^1.

Log Log plot ^{of the intrinsic} viscosity (cm’/g)

as a function of molecular weight. ^The point ^{marked Q}

is not used in the fitting ^{of formula} (9).

We could not check experimentally ^{the absence}

of shear rate dependence ^{for the} sample of molecular

weight MW ^{= 2.06 x 10’.} The reduced shear rate at the 0 temperature (cf. Table) ^{of this} sample ^{in the} microcapillary corresponds ^{to the onset of shear}

rate effect in [25], ^{but the} corresponding ^correction

is estimated to be less than 2 %, undetectable in ^our

experiment.

One can see in figure 1, that for this sample ^the

measured value of [11]0 [ = ^{360 :F 20} cm3/g, ⁱⁿ agree- ment with Toyo Soda’s value 340 cm3/g], ^{is smaller}

than that expected from formula (9). ^{This is related}

to the polydispersity ^{of the} sample, ^which viscosity averaged ^molecular weight M, ^is equal ^{to 1.6 x 107.}

3.1 CONCENTRATION DEPENDENCE AT T 0.

For

a given sample, ^{the two} capillaries with lowest shear rates (Cap. ^{1 and 2 cf.} Table) ^{lead to the same deter-}

mination of the specific viscosity _{il, ,P,} ^{which is} pro-

portional ^to the concentration for the whole range of T and M, ^{On the} contrary ^with capillary ³ (cf:

Table) ^{for T} 27°C and whatever the molecular

weight, the evaluation of flsp is different and presents

a second order concentration dependence (- Kri

term in formula (2)). ^{It seems} that this effect cannot be understood from an exclusively hydrodynamic point ^{of view.} Hydrodynamic interactions are pro-

portional ^{to the} occupied volume of the macromo-

lecules and decrease as the temperature is lowered This effect is related to the increase of thermody-

namic interactions as the temperature is lowered The phenomena ^{involved could} be, ^{for instance :}

-

a macromolecular adsorption ^on the wall of the capillary leading ^{to a} concentration variation of the sample.

-

a chain deformation induced by ^{the flow.}

Measurements presented ^{further on are not affect-}

ed by ^this problem, and there is no K. interaction coefficient for the entire _range of temperature;

thus flsp/C ^is always independent ^{of C.}

3.2 INTRINSIC VISCOSITY.

For MW ^{= 1.71 x 105}

and 6.77 ^x 106, flspl flspo is ^{plotted in figures 2 and 3}

versus temperature. ^{In each case the} experimental points ^{are obtained with two different concentra-}

tions and two capillaries of different shear ^rate.

Neither of these two parameters ^{has an influence on} the viscosity expansion ^factor a£.

Figure ^{4 is a linear} representation of the intrinsic

viscosity expansion ^factor a’ n ^{as a function} of i

for molecular weights ranging ^{from 8.2 x 104 to}

6.77 ^x 106. All the experimental points ^{lie on a}

master curve down to i fl = - ^{50. Our measure-}

ments are in agreement with those of reference [7]

Fig. ^2.

Viscosity expansion ^{factor versus} temperature (°C) ^{obtained with} MW ^{= 1. 71 x 105 and : +} cap. 1, ^{0 cap.} 2, C ⁼ 10-3 C*; ^x cap. 1, V cap. 2, ^{C = 10-2} C *. ^Characte-

ristics of the capillaries (cap.) ^are given ^{in the table.}

Fig. ^3.

Viscosity expansion ^{factor versus} temperature (OC) ^{obtained with} M,, ^{= 6.77 x 106 and : +} cap. 1,0 cap. 3, C ⁼ 3 x 10-4 C* ; ^x cap. 1, V cap. 3, C ⁼ 10-2 C*

Characteristics of the capillaries (cap.) ^are given ^{in the}

table.

Fig. ^4.

^Intrinsic viscosity expansion ^{factor as a function}

of the reduced temperature ^variable i Mw ^obtained

with 8.2 x 10’ M,., ^{6.77 x 106} (for symbols ^{see the} Table).

obtained with a molecular weight ^{of 3.19 x 10’ on a}

smaller _range at i fl. ^Thus an ^{is a} universal func- tion of the reduced variable ’t V MW’ whatever the molecular weight

From 0 down to ifl = - ^10, a’ ^decreases

linearly ^{as :}

The _power law (6) ^is tested in the representation

of figure ^{5. For - 60} ifl ^{- 40, the quantity} I a£ x i$ I ^{is a constant equal to 27 thus :}

This corresponds ^{to the} collapsed regime ^{with v =} 1/3.

Fig. ^5.

^{Plot of} oc’ x ^{I I V£ I versus the reduced tempe-}

rature variable t Jit. ^{The collapsed regime appears at}

T MW N - ⁴⁰ (for symbols ^{see the} Table).

Measurements performed ^{on the} sample ^{2.06 x 107}

are omitted in the former treatment which is _very sensitive to polydispersity. One ^can easily show ^that

the mean molecular weight M to ^{be used in} tle reduc- ed temperature ^variable r /m ^{is :}

-

near the 0 temperature I T IH I ^110,

-

far from the 0 temperature

Thus a polydispersity effect may explain ^{that the} experimental ^results obtained with Mw =t= ^{2.06 x} 10’ do not fit with the function of T.,,IM-, ^{shown in}

figure ^4.

As a definitive demonstration of the collapse regime, [ 1] ^is plotted, ⁱⁿ figure 6, ^versus the relative tempera-

Fig. ^6.

Log Log representation of the intrinsic viscosity (cm3/g) ^versus the relative temperature T ^for Mw ^{= 3.84} x 106,

6.77 x 106, ^{2.06 x 10’} (for symbols ^{see the} Table).

1770

ture T for MBB ^{= 3.84 x 106 and 6.77 x 10’. The}

intrinsic viscosity, inversely proportional ^{to the rela-}

tive temperature T, is molecular weight independent :

both molecular weight (3.84 ^{x 106 and 6.77 x} 106)

have same mean value of the intrinsic viscosity :

In the same figure ^{6 we} report measurements obtain- ed with Mw ^{= 2.06 x} 107, one can see that, ^{at a} given temperature (T ^{= 29.5} °C), both molecular

weights ^{6.77 x 106 and 2.07 x 107 have same mean}

value of intrinsic viscosity :.

This fact demonstrates that for i . JMW - ^{40 the}

chain is collapsed : ^intrinsic viscosity ^{does not} depend

on molecular weight ^and consequently ^on poly- dispersity ^{index, at a} given ^T.

4. Discussion

In the previous section, ^we have determined that for

! J Mw ;5 - ^{40 a collapsed} macromolecule is observ- ed. We discuss here how this collapsed regime ^is

reached using :

1) ^a Flory ^{mean field} approach ^with [ il] propor- tional to a global volume,

2) ^a scaling approach distinguishing ^{between sta-}

tic and dynamic lengths.

a3 experimental ^results below the 0 temperature

can be described phenomenologically by :

-

in the collapsed regime,

4.1 MEAN-FIELD APPROACH.

Near 0 and for T > 0 it is known [28, 16] ^{that the intrinsic} viscosity

measurements are in agreement ^{with a linear} expan- sion in z (a; ^{= 1 + 1.05 z) for : 0} z , 0.6 with :

z = 8 x 10 ’,r,.,IM-, ^{[29]. Such a} behaviour is well.

predicted by ^an original Flory equation (Eq. (4) ^with

y ⁼ 0) ^{in the context of the two parameter} theory

where a ⁼ 1 four 1 ⁼ 0. The expansion (10. a) ^{is an} experimental extension of this linear expansion ^to negative values of z valid till z 20138 x 10-2.

In order to prevent ^the instability of the chain at T 0 (in the theoretical description), ^{the three} body

interaction term is introduced, leading ^{to the modified} Flory equation (Eq. (4) with y =A 0).

For ^{i =} 0, ae is not equal ^{to 1 and} obeys ^the equa- tion :

The modified Flory equation ^{can be} expanded ^{as :}

-

for a & 1

Using ^formulae (10), (11), (12) ^{and an} identification of

cx ³ ₃

( Cxo) ^{with an, we determine an order of magnitude}

of y and oc,, :

This y ^value corresponds ^{to a smooth} shrinking ^{of the}

chain [19] ^{and is} compatible ^with experimental ^obser-

vation and with the measurements of reference [30].

But there is a shift A0 between the two definitions of the 0 temperature, ^{1 = 0 and a =} 1, (AO. M,- I -- 104),

which is molecular weight dependent. ^A0 may be due to an artifact of the mean field calculation [19].

4.2 SCALING APPROACH.

This theory ^considers

the asymptotic regimes. ^{In order to} describe the

cross over from 0 to good ^solvent regimes, ^{the blob}

model [17] has been introduced using the reduced temperature ^variable 1 jl

It has been suggested [17] that the intrinsic vis-

cosity ^{is :}

RG ^being the radius of gyration of the chain and RH

its hydrodynamic radius. This expression ^{is well}

verified in _very good ^solvents (cf. [17]). Considering

the radius of gyration expansion ^factor aG and the

hydrodynamic ^radius expansion ^factor aH, an ^may

be expressed ^{as :}

Below 0 we use the same approach. Figure ^{7 is a} plot ^of (X. au ^versus 1: JAi:, ^{where aG au values are}

interpolated ^from light scattering measurements of

RG ^{[7, 8] and} Ru [9] ^{on the} PS-cyclohexane system [31]. ^{In the} range of i fl(- ^{35 $} ifl 0)

where such measurements are available. We do not

find good ^{agreement between} a; ^{and aG au. In figure 7}

Fig. ^7.

Expansion ^{factor a’ as a} function of the reduced temperature ^variable i MW : ex;, ^{0 L-t3 p aG aH.}

a3 ^{is also compared to aH which seems to} fit better the evolution of ’7 . ^This discrepancy with formula

(14) although systematic may be related to experi-

mental difficulties in determining ^these magnitudes.

We noted in a previous paper [9] that relation (13)

involves a numerical factor which _may depend ^on

chain statistics and then _may have different ^values

in each asymptotic regime. The increase of this coefficient from the 0 to the collapsed regime may

explain the fact that L-4 3variations ^are smoother than

aG aH variations.

Conclusioa

Intrinsic viscosity measurements have been perform-

ed on polystyrene-cyclohexane system ^{for T 0.}

To perform measurements in the collapsed regime

the following conditions must be fulfilled :

-

high ^molecular weight samples (Mw Z ^{3.84 x} 106)

-

low concentration (CICO* ;: ^{5 x} 10-4).

Thus a precise viscosity measurement is required

Cj rr = ^{+ 10-4 . The intrinsic} viscosity expansion Q

-

/ ^{Y p} .__

factor scales with a single reduced variable ifl

whatever the molecular weight (8.2 ^x 10’ MW

6.77 x 106) from the 0 to the collapsed regime

(i MW - ^{40). In this} asymptotic regime, [il]-l

is independent of molecular weight and is then direc-

tly proportional ^to the internal concentration of a

chain of infinite molecular weight.

An index relating ^{size to molecular} weight v ⁼ 1/3

is derived at several temperatures through ^the asymp- totic behaviour [ q] - I T - ’ MW ^with Mw ^{= 3.84 x} 106, ^{6.77 x 106 and 2.06 x 107.}

Acknowledgments.

We would like to thank J. P. Carton, ^{P. G. de} Gennes, G. Jannink and H. Yu for fruitful discussions.

References and notes

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392.

[4] PRITCHARD, ^M. J., CAROLINE, D., Macromolecules 13

(1980) ^957.

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[6] SUN, ^S. T., NISHIO, I., SWISLOW, G., TANAKA, T., J. Chem. Phys. ⁷³ (1980) ^5971.

[7] MIYAKI, Y., FUJITA, H., Polymer ^{J. 13} (1981) ^749.

MIYAKI, Y., Thesis, ^Osaka University, Japan (1981).

[8] OYAMA, T., SHIOKAWA, K., BABA, K., Polymer ^{J. 13} (1981) ^167.

[9] PERZYNSKI, R., ADAM, M., DELSANTI, R., ^J. Physique

43 (1982) ^129.

[10] VIDAKOVIC, P., BONDELEZ, F., Macromolecules 17

(1984) 418.

VIDAKOVIC, P., Thesis, Université Paris VI, ^France (1983).

[11] WILLIAMS, C., BROCHARD, F., FRISCH, ^H. L., ^{Ann. Rev.}

Phys. ^{Chem. 32} (1981) ^433.

[12] ^{KRIGBAUM, W.} R., CARPENTER, ^D. K., ^J. Phys. ^Chem.

59 (1955) ^1166.

[13] YAMAMOTO, A., FUJII, M., TANAKA, G., YAMAKAWA, H., Polymer ^{J. 2} (1971) ^79.

[14] CUNIBERTI, C., BIANCHI, U., Polymer ¹⁵ (1974) ^346.

[15] See for instance : CHAMPETIER, G., Chimie Macromo- léculaire (Herman Paris) ^1972.

[16] YAMAKAWA, H., Modern Theory of Polymer ^Solutions (Harper ^{and Row, New} York) ^1971.

[17] WEILL, G., ^DES CLOIZEAUX, J., ^J. Physique ⁴⁰ (1979) ^99.

[18] ^DE GENNES, ^P. G., Scaling Concepts ⁱⁿ Polymer Physics (Ithaca ^{and New} York, ^{Cornell Univ.} Press) ^1979.

[19] ^DE GENNES, ^P. G., ^J. Physique ^{Lett. 36} (1975) ^L-55.

[20] WEBMAN, I., LEBOWITZ, ^J. L., KALOS, ^M. H., ^J. Phy- sique ⁴¹ (1980) ^579.

[21] Hagenbach corrections are neglected ^as leading ^to

maximal 03B4~sp/~sp ~ ^{5 x 10-3.}

[22] FERRY, J., Viscoelastic properties of polymers (J. Wiley

and Sons, New York) ^1980.

[23] EINAGA, Y., MIYAKI, Y., FUJITA, H., ^J. Polym. ^Sci.

Polym. Phys. ^{ed. 17} (1979) ^2103.

[24] STEPANEK, P., PERZYNSKI, R., DELSANTI, M., ADAM, M.,

to appear in Macromolecules.

1772

[25] SUZUKI, H., KOTAKA, T., INAGAKI, H., J. Chem. Phys.

51 (1969) ^1279.

[26] STRAZIELLE, C., BENOIT, H., Macromolecules 8 (1975)

203.

[27] This 03B8 shift would only ^affect by ¹⁰ % the coefficients in the 03B13~ expressions ^{which are} given ^without

error bars and rather as order of magnitude.

[28] TANAKA, G., Macromolecules 15 (1982) ^1028.

[29] ^{This z} expression is derived from reference [16] ^for- mula (39.6) ^with (39. 5), (40.13), R = ~6 RG ^and

RG ⁼ 0.29~Mw ^{10-8 cm from [13].} In this cal- culation we have supposed that PS decalin is

analogous ^{to PS} cyclohexane.

[30] ^DI MEGLIO, ^J. M., OBER, R., PAZ, L., TAUPIN, C.,

PINCUS, P., J. Physique ⁴⁴ (1983) ^1035.

[31] ^{We chose} measurements for which the 03C4~Mw ^conser-

vation is clearly established.