Temperature dependence of hydrodynamic lengths in the polystyrene-cyclohexane system

Temperature dependence ^of hydrodynamic lengths

in the polystyrene-cyclohexane system

M. Adam and M. Delsanti

DPh-G/PSRM, ^CEN Saclay, ^B.P. 2, 91190 Gif sur Yvette, ^France (Reçu ^{le 13} septembre 1979, ^{révisé le} 4 février, accepté ^{le 3}

^mars

1980)

Résumé. 2014 Nous présentons des résultats expérimentaux ^{sur le} comportement dynamique ^du système polystyrène- cyclohexane ^{en fonction de la} température

^au

voisinage ^du point 03B8 (0 ~ T 2014 03B8 30 °C). Il est trouvé que la

dépendance ^en température des rayons hydrodynamiques, RH ^et 03BEH, d’une chaine

en

régime ^{dilué et d’un blob}

^en

régime semi-dilué sont respectivement :

RH~ (T - 03B8) 0,034±0,005 03BEH ~ (T - 03B8)-0,33±0,04.

Nous montrons que

^ces

variations sont dues à la relative insensibilité des longueurs hydrodynamiques

^aux

^effets

de volume exclu. Les comportements observés semblent compatibles ^avec l’interprétation ^sur l’exposant dyna- mique apparent par Weill ^et des Cloizeaux.

Abstract.

²⁰¹⁴

We present experimental ^data

^on

^the dynamic temperature ^{behaviour of the} polystyrene-cyclo-

hexane system ^{in the} neighbourhood ^{of the 03B8} point (0 ~ T 2014 03B8 30 °C). It is found that the temperature depen-

dence of the hydrodynamic ^radii RH and 03BEH ^of

^a

^{chain in dilute} regime ^and

^a

^blob in semi-dilute regime

^{are res-}

pectively :

RH ~ (T - 03B8)0.034±0.005 03BEH ~ (T - 03B8)-0.33±0.04 .

We show that these variations

are

due to the relative insensitivity ^{of the} hydrodynamic lengths ^{to excluded volume} effects. The behaviours observed

seem

compatible with the Weill-des Cloizeaux explanation

^on

^the apparent dynamic exponent.

Classification

Physics ^Abstracts

05.40

^-

61.40

^-

66.90

^-

82.90

1. Introduction. - It has been shown, ^{in the} past few _years, that the dynamic properties of dilute and semi-dilute polystyrene ^{solutions can} be deduced from the static properties ^{if an effective} dynamical.

exponent Va, smaller than the static exponent ^{v, is} introduced [1]. The aim of this study ^{is to} compare static and dynamic temperature scaling ^{laws. The}

static laws, deduced from ^an analogy, ^{between criti-}

cal phenomena ^and polymer ^{systems, have been}

verified by ^{elastic neutron} scattering [2]. ^{Here we} report ^the dynamic température scaling ^{laws obtained}

from Rayleigh light scattering experiments ^on poly- styrene-cyclohexane system.

In the first part ^we recall the static scaling ^laws

and their extension to dynamic temperature ^laws obtained with the help ^of phenomenological argu- ments. Then, ^we report expérimental ^{results obtained}

in well defined temperature concentration domains.

The specifications of these domains have been done

using theory and static experimental ^results [2].

The data are fitted to power laws [23], ^{in order to}

compare those results with :

-

previous ^static [2] ^and dynamic ^results [1],

-

scaling ^laws theory [2,17].

-

the different explanations ^{on the} dynamic

exponent v. (effective ^{or real exponent} [3, 4]).

Finally ^we discuss the comparison ^{of our results}

with the recent Weill-des Cloizeaux explanation [3]

of an effective exponent Va.

2. Theoretical considérations.

^-

A polymer ^solu-

tion is considered [5] ^{as :}

-

dilute, ^if intermolecular interactions can be

neglected. ^{In this case} the characteristic length ^{is R}

the radius of gyration ^{of the chain ;}

- semi-dilute, ^if intermolecular interactions are

dominant. In a semi-dilute solution the chains inter- penetrate each other and the ^monomer concentra-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01980004107071300

tion [6] ^is bigger ^{than the} overlap concentration c*

defined by : ^{c* =-} NjR3, ^{where N} is the number of statistical elements in a chain ;

-

concentrated, ^{if the monomer} concentration is of the order of magnitude of the solvent concentra- tion. We will not consider them, ^here.

2.1 DILUTE REGIME.

^-

We consider an isolated chain of N statistical elements at a temperature T, above the 0 point. ^{The 0} point corresponds ^{to the} temperature ^where intramolecular interactions vanish.

The mean distance between two monomers, separated by ⁿ statistical elements, ^{has a} Gaussian behaviour if

n ne and an excluded volume behaviour if ⁿ > n,,.

The n, ^{value can} be derived from an extension of the well known Flory equation ^{of the} expansion

factor [8] :

where 1 is the length ^{of a} statistical element and ^v the excluded volume parameter. ^{The excluded volume} interactions are dominant when a > 1 or

which leads to :

using ^{a linear} approximation ^{in the} neighbourhood

of the 0 point, ^{v is} proportional ^{to the reduced} tempe-

rature ^r

⁼

(T - 0)/0 [7].

At a temperature ^T > 8, ^{one can visualize} [9, 10]

the chain as being ^formed by ^a succession of N/nC

Gaussian subchains of size :

Since excluded volume effects exist only ^between subchains, the radius of gyration R ^{of one isolated}

chain is equal ^{to :}

From expressions (1) ^and (2) ^{we obtain} [2] :

In the temperature ^{domain s} > N-l/2, this law is valid because the number of subchains is _very high (N/nC > 1). In the 0 domain ^z N- 1/2 the ^confor- mation of the chain is essentially Gaussian. The limit between these two behaviours is -r* = N-l/2 which will be called the cross-over temperature (see Fig. 1).

Now, ^we extend these results to the hydrodynamic

radius RH ^{defined as :}

where Do is the translational diffusion coefficients of the isolated chain and _’10 the viscosity of the solvent.

From an experimental point ^of view, it is known that :

Fig. ^1.

^-

Temperature concentration diagram : Region I) ^dilute system with excluded volume effect ; I’) ^{dilute 0} domain ; II) ^semi-

dilute system with excluded volume effect ; II’) semi-dilute 0 domain.

-

for a Gaussian chain (0 temperature) RHO ^and Re

are proportional [11],

. ^-

when excluded volume effects are present (good solvent) RH ^{is no} longer proportional ^{to R and} RH = ^{NVH 1} where vH ^{is an effective} dynamic expo- nent smaller than v [12].

For the dynamic behaviour, ^{we have} by analogy

with (3) :

The hydrodynamic ^radius, ’cH of a subchain (Gaus-

sian conformation) ^is proportional ^to rC. Taking ^the expressions (6), (2) ^and (1), ^{we have in the} tempe-

rature domain [13] r > -r* :

The value of the dynamic exponent vH, smaller than v,

implies ^{that the} hydrodynamic ^{radius must increase}

with the temperature ^more slowly than the radius of gyration. The thermal behaviour of RH ^{can be}

determined by quasi-elastic light scattering experi-

ments on a solution of finite ^monomer concentra- tion _c c*, ^{at a momentum transfer K such that}

KR ^1. The characteristic coherent time of the scattered light ^is [12, 14] :

D(c) ^{is the} diffusion coefficient of the coil at finite concentration. The experimental hydrodynamical ^I radius, RH(c), is deduced from the D(c) measurements from :

In the dilute regime, concentration dependence ^of

the diffusion coefficients, ^can be written as :

By extrapolation ^to infinite dilution (c

^-

0) ^we

obtain the hydrodynamic ^{radius of an} isolated chain.

The interaction parameter Q ^{is given} by [15, 6] :

m is the mass of monomer and v the specific ^volume

of the polymer ^{which is a constant in the} temperature

range investigated [16]. A2 is the second virial coeffi- cient and _gf the hydrodynamic interaction parameter of the friction coefficient (see appendix). ^{We demon-}

strate in the appendix that the variation of ç ^with

the temperature ^{is :}

where Bi, B2, B3 ^are temperature independent posi-

tive factors (see appendix). Difference between hydro- dynamic ^and geometric expontnt implies ^{that the} sign of ç ^{+ vm must} change ^{at a} finite, positive temperature so.

2.2 SEMI-DILUTE REGIME. - At a monomer con-

centration, c, higher ^{than the} overlap ^concentra-

tion c*, ^contact points exist between interpenetrated

chains. The _average distance between adjacent ^contact points, ç, ^is independent ^{of the} polymerization ^index

of the chains and decreases with concentration [5].

The semi-dilute solution at a temperature T > 0

can be considered as a non interacting assembly ^of

small chains of size ç, ^called blobs, having N(ç)

statistical elements. One can visualize one blob as a

succession of N(ç)jnc Gaussian subchains with excluded volume effects between them. Using ^an

argument ^{similar to the} argument developed ⁱⁿ

section 2.1, ^{we deduce :}

with the expressions of rC and ne ^{as a} function of the reduced temperature (1, 2) and the value of N (ç) :

we obtain [2] :

The size of the blob decreases with increasing tempe-

rature as it has been verified by ^elastic neutron scat-

tering experiments [2]. ^The expression (15) ^is meaning-

ful only ^if N(ç)/nc > ^1, this leads (using (1), (14)

and (15)) ^to the condition r » r** where T**

⁼

cl’.

In the 0 domain r « T**, the blob is essentially

Gaussian (see Fig. 1).

Let us consider now the temperature behaviour of the hydrodynamic length ÇH [17] ^{defined as :}

where D(ç) is the diffusion coefficients of the blob and _’10 the viscosity ^{of the} solvent. From light ^scat- tering experiments [1, 18] ^{it has} been found that ÇH

is not proportional to ç and scales with N(ç) ^{as :}

Using ^an argument analogous ^{to the one deve-} loped in the dilute regime ^we found that :

Using expressions (1), (2), (14) ^and (15) ^{we obtain :}

Since vH is smaller than v, the hydrodynamic length Çu

must vary with temperature ^more rapidly ^{than the}

static length.

The behaviour of ÇH ^{can be} determined from quasi-

elastic light scattering ^{on a} semi-dilute solution at ^c > c* and at a momentum transfer K such that K03BE « 1. The coherent characteristic time of the scattered light ^{is :}

from which we deduce the length ÇH by :

In the following section, ^we describe the experiments

which allow us to deduce the exponent values of the temperature ^laws.

3. Experknental procedure.

^-

^The dynamic lengths

are determined from spectral measurements of the

Rayleigh light ^scattered by cyclohexane-polystyrene

system, ^{in the 35 to 65 °C} range temperature, by light beating spectrometry.

Here we give ^the experimental conditions and the data processing used in order to determine 03BEH ^and RH,

we do not enter into the details of the light beating spectrometer which have been widely described in ref. [1].

3 .1 1 EXPERIMENTAL CONDITIONS. - The tempera-

ture regulation is achieved by ^a copper jacket regu- lated by ^a thermoresistance and ATNE monitor, within ± ^0.01 °C. The _copper jacket, in which the

optical ^cell containing ^{the solution takes} place, ^is

thermalized by ^a second chamber. The temperature is measured by ^{mean of a} thermocouple, ^{in contact}

with the cell, ^{within an} accuracy of ± ^{0.04 °C.}

To investigate the thermal behaviour of RH ^and 03BEH,

it is _necessary to carry ^out the experiments ^{in well}

defined regions. ^{To this} goal, ^{it is useful to locate}

the boundaries between the different regions, c*, s* and i**, given ^{in the} diagram (Fig. 1). ^{The cross-}

over temperature, ^{i* and} i**, ^are determined from

the temperature concentration diagram ^{drawn with}

results obtained by ^neutron scattering experiments

in P.S. cyclohexane system [2]. ^It is found that : 01* m 3

x

103 M- 1/2 in the dilute regime ,

ei ** = N 35 c in the semi-dilute régime. (21)

M is the molecular weight ^{of the} macromolecule,

c the monomer concentration [6] ⁱⁿ g/cm3. ^The

0 temperature ^{in the} polymer-solvent ^{system studied}

is equal ^{to 35 I)C} [19]. ^To define the dilute regime, ^it

is important ^to know the order of magnitude ^{of the} overlap concentration c*(t). We estimate that :

where cé ^and c*GS ^{are the overlap} concentrations at the 0 temperature ^{and in a} good solvent, respectively.

They ^{are obtained} using ^the following expression :

A is the Avogadro number, R; is the radius of gyration

found by intensity light scattering experiments [20].

The characteristics of the samples ^{used are} reported

in table I.

In order to observe macroscopic properties ^we

have performed ^the experiments ^{at :} KR gs ^{0.5 in} the dilute régime,

and

in the semi-dilute régime.

We use an homodyne spectrometer in the dilute

régime, ^{and an} heterodyne spectrometer ^with sepa- rated beams in the semi-dilute regime [1].

3.2 DATA PROCFSSING.

^-

As we deal with diffuse-

ing macroscopic process, the autocorrelation function of the scattered light intensity ^{has an} exponential profile ^{with a} characteristic time _Tc inversely pro-

portional ^{to K2.} This allows to determine a diffusion coefficient D

⁼

T;lfK2.

The momentum transfer K is related to the scat-

tering angle 03C8, ^{in the} solution, by the relation :

À (4 880 À) ^{is the} wavelength ^{of the} incident beam and n is the refractive index of the solution. K has been determined from measurements in air of the

scattering angle, taking ^{for the} cyclohexane ^refrac-

tive index [21] :

under these conditions, ^{we have obtained} diffusion coefficients with an expérimental accuracy of 0.5 %

in the dilute regime ^{and 1} % ^{in the} semi-dilute regime.

We have deduced RH and ÇH ^{from D} using expres- sions (9), (16) respectively ^{and a} temperature depen-

dence of the cyclohexane viscosity qo(T) established from a compilation ^of experimental ^results [22] :

the _accuracy on No(t) is better than ¹ %. ^This pro-

cessing gives RH(c --.> 0) ^and Ça ^{within an accuracy}

of 1.5 % ^{and 2} %, respectively.

4. Results and discussions 4.1 ^{DILUTE REGIME}

-

The study ^of RH 1 ^{as a} function of the concentra-

tion, ^{at a} given temperature, shows that RH 1 ^varies

linearly with concentration (Fig. 2). ^{From the} expe- rimental results and the expression [6] :

we can deduce I?ï 1(t) ^and tp(T) ^{defined in} (5) ^and (9).

4 .1.1 Hydrodynamic ^radius RH at finite ^dilution.

^-

The expansion ^factor Rai RH, ^varies by ^{less than 8} % (see ^Table II) ^{in the} temperature range investigated :

Assuming ^a power law [23], in the domain T/T* > ^1.4,

the best fit gives [24] :

The variation of RH/RHO, in the domain i/i* > 1.4

(domain ^{1 in} figure 1) ^{as a} function of 0-r is _repre-

Table I.

^-

Characteristics of ^the samples used where the symbols have their usual meaning. cGS*, ^{is calculated}

using of ^the expression (23) ^where Rp ^is the radius of polystyrene-dissolved ^{in benzene} [20].

Fig. ^2.

^-

Variation of the inverse of the apparent hydrodynamic

radius

as a

function of the

monomer

concentration, in the dilute

regime, ^{at T}

⁼

^{38.5 OC.} Representative

^error

bar is shown.

Table II.

^-

RH ^and ÇH experimental values obtained

at various temperature.

sented on a log-log ^{scale in} figure ^3. The aH exponent value is confirmed by ^the representation ^{used in} figure ^{4 :}

has a constant value for i/I* > 1.1, ^{but a} decreasing

value for i/i* 1, ^{which means that in the 0 domain} the results cannot be described by ^a power law. We

can note that the cross-over temperature i* coincides with the value determined from _eq. (21). The expo- nent aH found is much smaller than the exponent

a

⁼

0.2 predicted ^{for the} geometric expansion ; hydrodynamic ^AND GEOGMETRIC /LENGTH ^{are not} proper-

hydrodynamic and tional. This result is in geometric length are agreement ^{with ref.} not propor- [25] ^where

the ratio RJR is measured as a function of tempera-

ture in polystyrene-cyclohexane ^solution. RJR ^is

found to decrease as the temperature ^increases.

From the expression (7) ^{and the} experimental ^{value aH,}

we deduce an effective dynamic exponent vH

⁼

0.52.

The exponent value obtained is smaller than v(0.6)

and vH(o.55) obtained in _very good ^solvent [1, 12].

The _very low relative variation of RH compared ^to

the experimental accuracy does not allow us to rely

Fig. ^3.

^-

Variation of the hydrodynamie expansion ^factor

^{as a}

function of the temperature, Ot >- ^{10 0(;} (log-log scale).

^-

^Full

line is the best fit of the experimental points ;

^----

Dotted line represents ^the theoretical expansion ^{factor of} geometric length

which bas been calculated using arbitrary R/Ro

⁼⁼

RJR, at

0r

⁼

10 °C.

Fig. ^4.

^-

Verification of the température power law of the hydro- dynamic expansion factor, in dilute regime.

on the exponent ^value obtained from this experiment.

The important point is that the hydrodynamic

radius is relatively insensitive to the excluded volume effects compared ^{to the} geometric length.

4.1.2 Concentration dependence of ^the hydrody-

namic radius.

^-

In the temperature range investi-

gated (0 ^{°C Or 27} °C), ^the parameter

is negative and increases as the reduced temperature increases. The variation of Ç(i) ^with temperature ^is given ⁱⁿ figure ^5.

In the 0 domain ^T T*, the Q variation with the reduced température ^{can be} considered as being

linear. In the demain T > T*, experimental ^{data are} compared with calculated values obtained using

eq. (12) ^{with : :}

-

the exponent value found previously ^{vH = 0.52,}

-

and v

⁼

0.92 cm3/g ^found by densitometry [16].

The constant factor B2 ^and B3 ^are ajusted by ^means

of least square fit.

Fig. ^5.

^-

Variation with temperature of the absolute value of the interaction parameter 0. ^{Full line} corresponds ^to the theore- tical values obtained from expression (12).

The interaction parameter Ç and the excluded volume effects increase as the temperature ^increases.

Ç ^is negative ^{in the} neighbourhood ^{of the 0} point

and positive ^{for a} good ^solvent [1, 12, 26] ^{and it}

becomes equal ^{to zero at a} temperature ()to = 35°C

much higher than the 0 temperature [25]. ^This change

of sign ^{of the} parameter indicates that the thermo-

dynamic ^factor NA2 ^{is not} proportional ^{to the} hydro- dynamic parameter Qf. This fact confirms that R and RH ^{do not have the same} temperature dependence (see appendix).

4.2 SEMI-DILUTE REGIME.

^-

The measurement of

ÇH, ^{at a} given concentration (c = ⁴ Co*), ^{as a function}

of the temperature, shows that ÇH ^{decreases as the} temperature ^increases (see ^Table II). The results obtained in the region i/1** > 5.5 are plotted ^{on a} log-log ^{scale as a} function of the reduced tempera-

ture (Fig. 6). ^{It is} found that [24] :

Fig. ^6.

^-

Variation of the relative hydrodynamic length ÇH/ÇH6 in

semi-dilute regime

^{as a}

function of the temperature, 01 a 18.4 OC

(log-log scale). ^Full line is the best fit of the experimental points ;

^---

Dotted line is the variation of geometric length ^which

has been calculated using arbitrary (/(o

⁼

ÇH/ÇH6 at ^{0r = 18.4 OC.}

This exponent ^{value is} bigger ^{than the value}

b

⁼

0.25 ± ^0.01 determined by ^neutron scattering ⁱⁿ

the same temperature ^domain [2]. ^The hydrodynamic length 03BEH is ^not proportional ^{to the} geometric length ç.

In figure ^{7 we} present ^the experimental ^{results over}

the whole temperature range investigated. ^{We see}

that ÇHjÇH8 ^{has two different} behaviours, ^the quantity

the temperature r**exp ^{= 4} z**, where ÇH ^{reaches an} asymptotic behaviour, is in agreement ^{with the cor-} responding value obtained for 03BE by ^{elastic neutron} scattering experiments [2], but is much bigger ^than

the temperature i** determined from _eq. (21). Now, using expression (18) ^{and the} experimental ^value bH,

we found that the effective dynamic exponent vH is

equal ^{to 0.54. The} discrepancy between the exponent values found for the geometric length03BE ^{and the} hydrodynamic length ÇH ^{is due to the} insensitivity

of ÇH ^{to the} excluded volume effects.

Fig. ^7.

^-

Verification of the temperature power law of hydro- dynamic length in semi-dilute regime.

The different values of vH and v lead to opposite

effects on RH ^and ÇH’ RH ^{varies more} slowly ^{than R}

with temperature and ÇH ^{varies more} rapidly than j

with temperature.

4.3 DISCUSSION ON THE DYNAMIC EXPONENT. -

The existence of an effective dynamic exponent vH smaller than the static exponent ^{v allows us to} explain

the temperature ^{behaviour of the} dynamic quantities.

Let us relate our results with the Weill-des Cloizeaux [3a] (W.D.C.) interpretation ^{of the} appa- rent exponent ^vH.

They (W.D.C.) calculate the radius of gyration (R - Ny) ^{and the} hydrodynamic ^radius (RH - NVH)

of a chain taking ^{as for} hypothesis ^{the Gaussian}

conformation of the chain ^at shorter distances, they

determine the effective exponents v(Nlnc) ^and vu(Njne).

At a low Nln,, ^value, vH is found to be smaller than v.

As Njne ^{increases v} reaches its asymptotic ^value [27]

(0.6) ^more rapidly ^than vH.

This treatment can be extended to the calculation of the effective temperature exponents ^{if one notes :} - ne is proportional ^{to t - 2 in the} neighbourhood of the 0 point,

-

the blob is analogous ^{to an isolated chain of} size ç ^with N (ç)- cç 3 statistical elements.

In figure ^{8 we} give the variation of the effective temperature exponents a, aH, b ^and bH ^{as a function}

of J Njne. In the dilute regime, ^{for the} temperature

range 1 T/T* ⁴ (or ¹ BNlnc 4) ^{where the} experiments ^{have been} performed, the effective tem-

perature exponent varies between 0 and 0.1; ^the

mean value correspond roughly ^{to the} exponent determined experimentally (aH

⁼

0.034). ^From figure 8, [28] ^we observe that the exponent - bH

decreases _very rapidly and reaches a minimum value

(-bH= -0.3) ^at JN(ç)/nc = 2.5 and then increases very slowly ^with N(ç)jne (- bH

^{= -}

^{0.29 for}

J N(ç)jne

⁼

5). Experiments, in semi-dilute regime,

have been perforrned ^{in the} temperature range 1 T/T*exp* ^{3 ’ which} corresponds, making ^{an over} estimation, ^{to 1} JN(ç)jne ^{3. In this tempera-}

ture domain bH ^is approximately ^{a constant} bH= ^0.3

which must be compared ^to the value determined

experimentally bH

⁼

^0.33 [29]. ^{We can note} that, ⁱⁿ order to determine vh ^from bH ^{we must know thé}

v corresponding value, ^because using ^the asymptotic

value v

⁼

0.6 one over estimates the exponent value _vH.

This discussion leads to the conclusion that the

explanation given by Weill and des Cloizeaux is in

qualitative agreement ^{with our} experimental ^results.

The main problem ^aroused by W.D.C. calculation

Fig. ^8.

^-

Variation of the temperature exponents,

^a

^and aH in dilute regime, b and bH ⁱⁿ semi-dilute régime

^{as a}

function of

Q ^{which is equal to i/1* in dilute régime and} approximately equal ^to i/i** in semi-dilute regime.

lies in its basic assumption : does the local Gaussian conformation exist ?

This assumption ^is verified in a 0 solvent ; ^but there is no experimental evidence of its validity ^{in a} good ^solvent.

By ^{elastic neutron} scattering ^the size rc ^{of the}

Gaussian conformation, ⁱⁿ polystyrene-cyclohexane

solution near 0 temperature, ^has been measured [30].

rc decreases as the temperature increases and becomes

equal ^to the statistical length (1

⁼

¹⁰ À) ^when

Since benzene, ^{at room} temperature ^{is a better} solvent than cyclohexane ^{at 57 °C for} polystyrene (Rbenzene > Rcyclohexane ⁵⁷⁰⁾ the local Gaussian confor- mation does not exist in the system polystyrene-

benzene. This observation casts some doubt on the

validity of W.D.C. calculation and we believe that direct experimental confirmations are indispensable.

One of these, ^could be, ^to find some polymer ^solvent

systems where this cross-over effect is negligible, ⁱⁿ

order to verify ^that vH could indeed reach the _asymp- totic value (0.6).

5. Conclusion.

^-

From these experiments, ^{done in}

the neighbourhood of the 0 temperature ^{on PS-} cyclohexane system, ^we conclude : that the hydro- dynamic ^radius RH ^{of a chain is} relatively insensitive to changes ^{in the} segment distribution arising ^from

excluded volume effects [31]. This conclusion can be

extended, ^{in the} semi-dilute regime, ^{to the} hydro- dynamic length ^{of a blob} 03BEH.

The concentration and molecular weight depen-

dence Of 03BEH ^and RH, ^{at the 0} temperature, ^{have not} been verified but the temperature exponents found, suggest that, 03BEH ^{c-’ and} RH - ^{Mû.5. This last} point ^{is in} agreement ^with experiments ^{done at the}

0 point [11]. Our results are in qualitative agreement with an explanation ^{of anomalous} dynamic expo- nents by cross-over effects. But those experimental

results (Table II) ^allow any theoretical approach ^on

the problem ^of dynamic exponents, ^to be verified.

Acknowledgments.

^-

The authors wish to thank J. P. Cotton, ^C. Bagnuls, ^{J. des} Cloizeaux, ^{G. Weill} and G. Jannink for hepful discussions ; ^{F. Geiessler} and A. M. Heicht for their advices regarding ^various experimental aspects. ^{We thank M. Daoud for} sending ^his preprint.

Appendix.

^-

^{Let us} recall the expression ^{of the}

interaction parameter ç [6, 15] :

^.

Here we relate, the second virial coefficients A 2 ^and

the hydrodynamic interactions parameter Qf, ^to the

macromolecular parameters ^{in order to} calculate the thermal behaviour of e.

In the domain i > i*, A 2 is proportional ^{to the}

volume occupied by ^{one chain} [8, 5], ^with eq. (4),

we have :

^.

In the 0 domain i i*, A2 ^is proportional ^{to the}

excluded volume parameter [8] ^{which is a linear}

function of the reduced temperature

Q f is defined [6, 15] by :

where fo ^and f (c) ^are the frictional coefficients of a

chain at infinite and finite dilution, respectively. ^To

evaluate _Qf let us select a chain amongst ^{the other} and détermine its effective friction coeffcient f (c) :

the additional friction coefficients bf(c) ^{is due to the} macroscopic incrément of the viscosity bn ^{due to the}

presence of the other chains :

03B2 ^{is a} numerical factor which is no longer equal ^{to 6 7c}

since the size of the selected chain is equal ^{the size}

of the other chains [32]. &, is related by definition [6, 8]

to the intrinsic viscosity M by :

Using ^the expressions (A . 4)-(A . 7) ^{we obtain :}

It has been shown [33] ^{that :}

where _il (characteristic ^{time of the} first mode of the

chain) is the time required by ^{a chain to diffuse on}

a distance equal ^{to its size} [34] R :

Using (A. 8), (A. 9) ^and (A 10) ^{we find that}

In the domain i > T* using the thermal behaviour

proposed ^for RH, ^R, A2 ^{and the} expression ^of e

(eqs. ^(4), (7), (A . 2) ^and (A 1», ^{we have :}

In the 0 domaine 1:*, ^a temperature expansion

can be made :

Factors BI, B2, B3 ^{and v are} independent ^{of the}

reduced temperature. ^The specific ^{volume v can be}

considered as a constant in the temperature range

experimentally investigated [16]. Bi ^is proportional

to the intrinsic viscosity (positive factor), ^{thus at the}

0 point e ⁺ vm is negative. B2 ^and B3 ^are positive

and respectively proportional ^{to N3Y-l and to} N"H-".

The reduced temperature io ^at which e ^{+ vm vanishes}

is molecular weight dependent (To - N-1/2).

Références and notes

[1] ADAM, M., DELSANTI, M., Macromolecules 10 (1977) ^1229.

DELSANTI, M., Thesis, Université de Paris Sud (1978).

[2] DAOUD, M., JANNINK, G., ^J. Physique ³⁷ (1976) ^973.

COTTON, ^J. P., NIERLICH, N., BOUÉ, F., DAOUD, M., FARNOUX, B., JANNINK, G., DUPLESSIX, ^{R. and} PICOT, C., ^{J. Chem.}

Phys. ⁶⁵ (1976) ^1101.

[3a] WEILL, ^{G. and}

^DES

CLOIZEAUX, J., ^J. Physique ⁴⁰ (1979) ^99.

[3b] AKCASU, ^A. Z., HAN, ^C. C., Macromolecules 12 (1979) ^276.

[4] JASNOW, D., MOORE, ^M. A., ^J. Physique ^{Lett. 38} (1977) ^L-467.

AL-NOAIMI, ^G. F., MARTINEZ-MEKLER, ^G. C., WILSON, ^C. A.,

J. Physique ^{Lett. 39} (1978) ^L-373.

[5] DAOUD, M., COTTON, ^J. P., FARNOUX, B., JANNINK, G., SARMA, G., BENOIT, H., DUPLESSIX, R., PICOT, C.,

^DE

GENNES,

P. G., Macromolecules 8 (1975) ^804.

[6] ^The

^monomer

concentration, c, in section 2 and appendix ^is expressed ⁱⁿ

^monomer

^number by unity ^volume (cm-3),

in the section 3 and 4 is expressed ⁱⁿ

monomer mass

by unity ^volume (g/cm3). ^{If A}

⁼

^{constant x B and the}

constant is dimensionless,

^we

will write A ~ B. A ~ B

means

A proportional ^{to B.}

[7] This definition of 03C4 is different from the

one

introduced by Flory [8]

^03C4

= (T-03B8/T) ^However, in the small values of 03C4

here investigated ^{there is}

^no

appreciable numerical diffe-

rence

between the ^two definitions.

[8] FLORY, ^J. P., Principles of Polymer Chemistry (Cornell) ^1953.

[9] FARNOUX, B., BOUÉ, F., COTTON, ^J. P., DAOUD, M., JANNINK, G., NIERLICH, N.,

^DE

GENNES, ^P. G., ^J. Physique ³⁹ (1978)

77.

[10] DAOUD, M., Thesis, Université de Paris VI (1977).

[11] KING, T. A., KNOX, A., LEE, ^{W. I. and} MCADAM, ^{J. D.} G., Polymer ¹⁴ (1973) ^151.

FROST, ^R. A., CAROLINE, D., Macromolecules 10 (1977) ^616.

[12] ADAM, M., DELSANTI, M., ^J. Physique ³⁷ (1976) ^1045.

[13] ^{DAOUD, M. and JANNINK, G., J.} Physique ³⁹ (1978) ^331.

[14] See for instance :

CUMMINS, H. Z., PIKE, ^E. R., Photon correlation and light beating spectroscopy (Plenum press, New York) ^1973.

[15] ^{YAMAKAWA, H., Modern} theory of polymer ^solutions (Harper

and Row, ^New York) ^1971.

[16] FRANÇOIS, ^J., CANDAU, F., BENOIT, H., Polymer ¹⁵ (1974) ^618.

[17] ^DE GENNES, ^P. G., Macromolecules 9 (1976) ^587.

[18] ADAM, M., DELSANTI, ^{M. and} JANNINK, G., ^J. Physique ^Lett.

37 (1976) ^L-53.

[19] STRAZIELLE, C. and BENOIT, H., Macromolecules 8 (1975) ^203.

[20] DECKER, D., Thesis, Strasbourg (1968).

[21] TIMMERMANS, J., Physics ^{chemical constants} of pure organic compounds (Elsevier Amsterdam) ^1950.

[22] Lange’s ^Handbook of chemistry, ^{J. A. Dean} (Mc ^{Graw Hill} book Cy) ^1973.

Handbook of chemistry ^and physics, 59th Edition (RC Weast

CRC Press).

BRUNET, J. and GUBBINS, ^K. E., Trans. Faraday ^{Soc. 65} (1969)

1255.

Polymer Handbook, Brandrup, J., Immergut, ^{E. H.} (1975).

[23] ^{Since the} analytical expression proposed ^{in ref.} [3b] ^{does not}

allow the authors to obtain

a

quantitative agreement We fit

our

data by power laws.

[24] ^A variation of 1 °C in the 03B8 value give

^no

appreciable change in the exponent ^value (6 %).

[25] ALLEN, G., VASUDEVAN, P., HAWKINS, ^E. Y., KING, T. A., J.

Chem. Soc. Faraday Trans. II 73 (1977) ^449.

[26] BAILEY, D., KING, T. A., PINDER, ^D. N., ^Chem. Phys. 12 (1976)

161.

KING, T. A., KNOX, A., MCADAM, ^{J. D.} G., Polymer ¹⁴ (1973)

293.

[27] ^A

more

accurate value for

03BD

is 0.588 (see ^ref. [4]).

[28] After this paper

^was

written,

^we

^received

^a

preprint by ^M.

Daoud giving similar theoretical results

on

the thermal behaviour in semi-dilute range.

[29] ^One

^can

remark that the maximum theoretical value of bH (using

^{03BD =}

0.6) is smaller than the experimental ^value.

[30] FARNOUX, B., BOUÉ, F., COTTON, J. P., DAOUD, M., JANNINK, G., NIERLICH, M.,

^DE

GENNES, ^P. G., ^J. Physique ³⁹ (1978) ^77.

[31] KRIGBAUM, ^{W. R. and} CARPENTER, D. K., J. Phys. ^{Chem. 59} (1955) ^1166.

[32] LI, J. C. M., CHANG, P., ^J. Chem. Phys. ²³ (1954) ^518.

[33] FERRY, ^J. D., Viscoelastic properties of polymers (Wiley, ^New York) ^1961.

[34] LANDAU, L. D., LIFSHITZ, ^E. M., Mécanique des fluides (Mir

Moscou) ^1961.