Viscosity of semi-dilute polymer solutions
M. 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>
HAL Id: jpa-00209425
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Submitted on 1 Jan 1982
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J.
Physique 43 (1982) 549-557
MARS
1982,
549
Classification
Physics Abstracts
46.30J
-
51.20
61.40K - 82.90
-
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
(c*
présentons des résultats
de viscosité obtenus dans des solutions semi-diluées de
polymères
10%).
c
La variation de la viscosité est indépendante de la température de transition vitreuse du polymère et compatible
coefficient de frottement du monomère proportionnel à la viscosité du solvant.
En fonction de la concentration dans différents systèmes, les relations suivantes sont obtenues :
PIB-toluene, ~r ~ c4,1, qui est proche de la théorie de la viscosité de de Gennes,
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 analogue
à 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 :
avec un
2014
2014
~r ~ M3,1w
en
accord
avec
le modèle de
reptation.
Toutefois, il semble que les lois d’échelle
tive
en
température ne peuvent expliquer la décroissance de la viscosité rela-
lorsque la température croit.
We report viscosity measurements on semi-dilute solutions (c*
c
Abstract.
10 %).
The viscosity variation is independent of the glass transition temperature of the undiluted polymer and consistent
with 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 reptation
and mean field theory of elasticity.
With molecular weight we observe :
2014
~r ~
in agreement with reptation model.
However temperature scaling laws do not seem to be
dence of the viscosity remains open question.
Zero shear viscosity of polymer
Introduction.
solutions and melts have been intensively studied for
many years. A good review article on the subject was
written 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-dilute
solutions. A semi-dilute solution may be considered
as a transient network, the monomer concentration c
being higher than the overlap concentration c*
(c* = N/R3 [3]) but much lower than the solvent
concentration. In order to be able to compare theory
-
M3.1w
applicable.
The
interpretation
of the temperature
depen-
experiments we have performed some new
experiments on very high molecular weight polymer.
First, we review de Gennes’ theory of semi-dilute
(S.D.) polymer solutions. Then we define the experimental conditions at which the experiments were
performed. Finally, we present and discuss the experimental results obtained in both, good and 0 solvents.
1. Theory.
The main term in the expression for
the viscosity 11 is the product of the longest relaxation
time TR and the shear elastic modulus E of the tranwith
-
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 E
and TR from de Gennes’ theory in the simplest case
of an athermal solvent (Flory-Huggins parameter
z = 0).
The main
to consider
a
Following rubber elasticity theory (7), the elastic
modulus of the transient network is proportional to
the volume number density of strands c/g. Using the
expressions (2) and (4) we obtain : E -- kT/ç3 thus :
hypothesis of the de Gennes’ theory is
chain (in a S.D. solution) as formed by
succession of subchains of mean size ç, each subchain having g monomers.
Inside a subchain, the g monomers are subject to
excluded volume effects and to hydrodynamic interactions. Such interactions do not take place between
two subchains; ç is a screening length for both excluded volume and hydrodynamic effects.
Edwards [4] and de Gennes [5] imagine that the
motion of a particular chain is a reptation in a fictitious tube, defined by all the surrounded chains. The
diameter of this tube is the screening length ç :
a
where I is an effective length per monomer
The length of the tube is :
From the expressions of T R
for the viscosity [10] :
monomers
in
a
since a S.D. solution is a closely packed system of
subchains units.
The time TR necessary for a complete renewal of
the tube is the time required for chain to reptate the
length of the tube [6] :
we
obtain
Using the most accurate experimental value of
[8], which corresponds to the prediction of the
vector model [9],
one
chain and
and E (9)
or
v
n
finds :
We have considered above the case of an athermal
solvent (x
0, v 0.588) where only one characteristic length ç exists. The quality of the solvent
should have no influence on the viscosity molecular
weight exponent, but it must influence strongly the
concentration dependence. One may remark that,
since the length ç decreases when the quality of the
solvent increases (Ref [2], p. 119, 1 l, 12), the viscosity
given by equation (10) must increase.
In the case of a 0 solvent [13] the theory of the viscosity is much more difficult because there exist two
=
where N is the number of
(8)
=
lengths [14] :
-
where Dt is the diffusion coefficient of the Rouse
chain along its tube. Thus :
-
tive
where qo is the solvent
viscosity.
Combining equations (3), (5)
and
(6),
we
the correlation length, which is the screening
of the density-density correlation function :
length
and the
mean
distance between two
consecu-
entanglements :
have :
is : how the shear modulus and the
time depend on ç and Ç2. In secrelaxation
longest
tion 3.2, we will discuss our experimental results
obtained in 0 conditions and compared them with
different theoretical possibilities.
The
thus :
problem
The principle of the
2. Experimental procedure.
viscometer was given in our earlier paper [15]. We
measure the viscous force F to which a sphere (radius
5 x 10-2 cm) is submitted when the fluid is
r
-
Substituting the expressions (4) and (2) for g and ç
in the expression of TR (7) we have :
=
displaced at a velocity
v :
551
Characteristics of polymer solutions studied.
Table I.
the
transition temperature of the undiluted
is
glass
Tg
polymer and X the Flory-Huggins parameter. Indexes
v and t refere to X deduced from intrinsic viscosity and
thermodynamic measurements, respectively.
-
the ratio F/v as a function of v and verify
has
a constant value, thus we determine the
F/v
zero shear viscosity of the sample.
We have made various improvements [16] upon
the viscometer specifications as compared to reference [15]. The force maintaining the sphere in place
is transmitted magnetically and we have improved the
feedback control of the current driving the field
coils. This current is of the order of 1 mA and it is
now measured to within 1 %. The sample flow is
we measure
that
driven by a stepmotor and its velocity is measured to an accuracy better than 0.1 %.
We made an absolute calibration of the apparatus
by measuring a standard silicon oil (q 50 P at
25 OC) and our new overall precision is 2 %.
From the measured viscosity, we have deduced the
relative viscosity defined as the ratio of polymer solution viscosity to that of the solvent [17], given in appendix 2.
Two kinds of polymer were used : poly-isobutylene
and polystyrene in three different solvents whose
characteristics are given in table I.
In figure 1 we plot the different concentrations of
the samples at which the viscosity was measured. We
represent also the variation of the overlap concentration c* as a function of molecular weight. As usual [12,
15,18] we use for c* :
now
=
where A is Avogadro’s number, p is the density of the
solvent, and RG is the radius of gyration measured by
light scattering [19]. For comparison, we plot (dashed
line) the value of cc, for PS system :
where cc (g/g) define the boundary between the entangled and non entangled regime [20]. We can note that :
the cc,
M-1, in good solvent cannot be justified 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 &#x3E; 106), we find no significant range of concentration where the system is semi-dilute but not
entangled [20].
3 .1
3. Experimental results and discussions.
3.1.1 Concentration dependence.
GOOD SOLVENT.
In the PS-benzene and PIB-cyclohexane systems,
the dependence of relative viscosity on concentration
is, at 32 OC, identical for both systems :
-
-
-
the exponent decreases sliat 64 OC, the expothe
increases,
ghtly
temperature
nent value is 4.4 + 0.1. This has to be checked on
PS-benzene at high temperature.
For PIB-toluene the viscosity, for a given concentration, is much lower than for PIB-cyclohexane :
For
PIB-cyclohexane
as
Variation of the overlap concentration c* as a
function of molecular weight M.. + - + PIB and .--8
PS samples on which concentration exponent values are
determined. The numbers on each line represent the Mw/Mn
PS-benzene c* values;
values. a) Good solvent systems.
.... PIB-cyclohexane c* values; ------ cc deduced
from [27]. b) 6 solvent systems.
PS-cyclohexane (35 OC);
· · · · PIB-benzene (25 °Q ;
Cc
Fig.
1.
-
-
-
------
552
4. - Variation of the relative viscosity as function of temperature for a PIB-cyclohexane sample
(c = 9.3 x 10- 2 g/g).
Fig.
Fig. 2. Relative viscosity at 32°C as a function of
24 x
centration in 0 PS-benzene samples (MW
and + PIB-cyclohexane samples (MW =1.17 x 106)
-
=
In the PIB-toluene system, the concentration
dence of the relative viscosity is :
con-
106)
[41].
depen-
In figure 5 we present the variation of r as a function of concentration for different systems (some
numerical values of q, as a function of temperature
are given in appendix 3).
In a dilute regime with short chains at high concen4 x 104)
tration (c
9 x 10 -2 g/g, c/c*
0.6, MW
the relative viscosity is very small and within experimental accuracy, independent of the temperature
(20 °C T 60 OC), see figure 5. Even with 9 %
solution concentration, the viscosity dependence on
temperature is that of the solvent. This proves that
for a monomer concentration less than 10 %, the
monomer friction is proportional to the viscosity of
the solvent (monomer-monomer friction can be
=
=
in agreement with reference [21]. The concentration
exponent, close to theoretical value of 3.95, is independent of the temperature.
neglected).
In figure 5,
=
observes that :
PS-benzene and PIB-cyclohexane systems present the same variation of r with concentration,
one
-
Fig. 3.
ples (M,
-
Relative
1.17 x
viscosity
106)
=
as a
at 64 °C in PIB-toluene sam-
function of concentration
[41].
The main effect,
3.1.2 Temperature dependence.
observed on S.D. solutions, PS-benzene and PIBcyclohexane, is that the relative viscosity q, decreases
as we increase the temperature.
An example of the decrease of the viscosity as a
function of temperature is given in figure 4. This
variation can be considered as linear and parameterizing it as :
-
Fig. 5. Variation of r (see text) as a function of the
concentration for different good systems. + PIB-cyclohexane ; * PIB-toluene; 0 PS-benzene, M, 7.1 x 106 ;
-
=
A
c
we
find : F
=
9
x
10- 3 in the case of figure 4.
PS-benzene, MW =2 x l(f;00 PS-benzene, Mv =4 x 104,
C - 0.6.
553
r
being independent
of the molecular
weight.
This
temperature 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 very
small compared to those obtained on PIB-cyclohexane.
When T increases, glgt and 1, increase and decrease,
respectively. This seems to justify that in the case of
PS-benzene and PIB-cyclohexane :
-
Qualitative interpretation of the results. polymeric systems studied (x = 0.44) the
effective length I is much larger than the size of a
3.1. 3
In the
monomer a.
Let us suppose that :
at very short distance
i)
rigid [22, 23], m being
affected by rigidity,
ii) at larger distance (r
r (r ma), the chain is
the number of monomers
&#x3E; ma) the subchain ç has a
Gaussian conformation if m g
gt. The excluded
volume effects are dominant if g &#x3E; g, [24].
In the range of molecular weight studied, we have
low values of glg, in the semi-dilute range. For instance,
in a PS-benzene solution, at room temperature, gt
corresponds to a molecular weight M. of = 104 [25].
By comparison of diffusion coefficient at zero concentration and cooperative diffusion coefficient in semidilute solution [18], having a monomer concentration
of 2 % c 10 % the corresponding Mg values are :
c = 10 %, the ratio g/gt is of the order of
In other words, at room temperature and for
c
10 %, the effective value of the exponent v
has not reached its asymptotic value.
Using the assumptions i) and ii) the expressions
glgt (A. 5) and r¡r (A. 6) are developed in appendix 1.
In those expressions T is a constant because the systems
studied (PS-benzene and PIB-cyclohexane) have very
low 0 temperatures [7] and small change near room
temperature does not affect the quality of solvents
(T const.). If we suppose that m is inversely proportional to the temperature we obtain the following
Thus, for
unity.
2%
=
temperature dependences :
independent
of the molecular
weight (see
Fig. 5),
-
In the systems studied (x = 0.44) the concentration exponent values found are larger than the theoretical value developed in section 1 for an athermal
solvent. In the systems PS-benzene and PIB-cyclohexane, the relative viscosity decreases as we increase
the temperature.
If the first result can be interpreted as due to high
x values, the second result cannot be interpreted
using usual temperature scaling laws [2, 11, 12]
which predict a thermal effect opposite to that observed. Indeed increasing the temperature, the quality of
the solvent increases and thus the viscosity should
increase (see section 1).
F is
-
the viscosity and the experimental concentration
as T increases.
exponent decrease
In the case of PIB-toluene the independence of the
relative viscosity with temperature and the low value
of concentration exponent could be due to high
flexibility of the polymer in this system giving a high
value of g/gt.
This agreement is only qualitative, we cannot go
further because the effective v is an unknown function
of concentration and temperature.
This assumption (that v has not reached its asymptotic 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 concentration range (0.5 %
c
5 %) lower than that of our
viscosity measurements, hence in a higher range of
gIg".
Self-diffusion measurements on PS-benzene soluc
tions are reported in reference [27] (2 %
20 Y.),
they give
the relation : Ds L-- R 2/ TR with R 2~ C- 1/4
and E - c2.2s (corresponding to v
0.6) leads to
in
This
C3.7.
agreement with
exponent (3.7),
q de Gennes’ prediction, is much lower than our 4.6
concentration exponent. Is the concentration exponent
of the self diffusion coefficient very sensitive to the
effective 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 field
theory. However this theory leads to a viscosity
concentration exponent of 3.5, smaller than the good
solvent exponent (3.95) and thus in contradiction with
our experimental results.
In short, the value of the concentration exponent
may, be interpreted as due to a non athermal solvent
polymer system. However, the decrease of the relative
viscosity with increasing temperature could not be
understood in terms of the classical temperature
dependence of ç, but in terms of local rigidity.
Using
=
3.2 0 SOLVENT :: PS-CYCLOHEXANE, PIB-BENZENE
The viscosity of S.D. solutions, at
SOLUTIONS.
0 temperature and at a given concentration is not
very different from that of S.D. solutions in good
solvents. This contrasts with what is observed in dilute
solutions, (Ref [29]). On both our systems, when we
increase the temperature from the 0 point, (T 2013 0 = 160)
the relative viscosity, for a 9 % monomer concentration, decreases by a factor 1.6.
-
554
Table II.
-
Different expressions of the viscosity
combining the expressions of E
ETR obtained
11
and T R ( 1 4 and 15).
=
of contact of two monomers, belonging
to a Gaussian chain, because the coils are real and
cannot cross each other.
probability
6.
Relative viscosity as a function of concentration
7 x 106),
at 6 temperature 0 PS-cyclohexane (M,,
T
1.17 x 106 ), T =25 °C
35 °C, + PIB-benzene (M,,,
[41], the arrows represent c* values.
Fig.
-
=
=
=
In both cases, PS-cyclohexane (35 °C) and PIBbenzene (25°C) the relative viscosity varies with
concentration as :
In figure 6 the arrows represent the c* values on both
systems. As soon as c &#x3E; c*, the experimental points,
on log log scale, lie on a straight line of slope 5.
Starting
from these
experimental
results
(1,, - c5)
to examine the contribution of the two characteristic lengths which are present in the system, at the 0
we want
temperature.
The first length is the mean distance between two
consecutive entanglements or contacts :
longest relaxation time (TR - N 3 cl) cora screening length proportional to c-’.
In that case the fictitious tube has a length :
-
The
responds
to
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 because
the pair interaction vanishes. In order to define a
correlation 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 rationalized
with the usual temperature dependence of the characteristic length in S.D. solutions. Using this temperature
dependence [11, 12], we find a relative viscosity which
increases with temperature.
3. 3 MOLECULAR
using
this distance it
The second
density-density
was
found
[14, 30] :
length is the correlation
correlation function :
WEIGHT DEPENDENCE IN PS-BENThe polydispersity of PS samples
makes the determination of the value of the molecular
weight exponent very delicate. We have used the best
commercial samples available.
The molecular weight exponent value is independent
of the quality of the solvent, thus of the concentration
exponent value (see section 3.1.1). Taking the experimental value r ’" C4.6 for PS-benzene samples at
32 OC we find :
ZENE SOLUTIONS.
length of
ç ’" c- I corresponds to the expression (2) of j
where v is set equal to 1/2. Using v
1/2 in the expressions of the shear modulus (9) and of the longest
relaxation time (8) one find :
-
=
with
from Toyo Soda company, having a
3.84 x 10’
of 1.04 and 1.14 for M,
6.77 x 106, respectively; and,
samples
polydispersity
Combining the expressions of TR and E given, the
different resulting viscosity expressions are reported
in table II.
Thus the only way to match the
results is to have :
Mw
=
experimental
E - c2
Shear modulus
relaxation
time
TR - N 3 C3 .
Longest
the elastic modulus corresponds to
Physically :
-
and
=
with
samples from
dispersity of 1.15
pressure chemical having a poly4.1 x 106 and
for both Mw
=
7.1 x 106.
This exponent value is very sensitive to
polydisper-
,
555
d) The viscosity of the PS-benzene and PIB-cyclohexane seems to be influenced by local conformation.
e) The viscosity at the 0 temperature is much more
complex than in good solvent scaling laws do not
seem 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 the
relative viscosity could be interpreted using de
Gennes’ theory of semi-dilute solutions. The problem
posed by these experiments is the decrease of the
viscosity when the temperature increases. The classical
temperature dependence of the characteristic length
-
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
weight Mv, we find instead of 3. l,
average molecular
exponent value
of 3.5. There is no reason to use the Mv mean value.
Making the assumption that the mean longest relaxation time of a polydisperse sample is the weight average
of TR [6], since the elastic modulus is independent of the
molecular weight, we have :
an
could not account for this variation.
The authors gratefully thank
J. P. Cohen Addad for the Precious PIB-sample;
P. G. de Gennes, P. Pincus and R. Ball for helpful and
Acknowledgments.
-
stimulating discussions.
Appendix
Let
us
mers a
suppose that
local
a
1
chain follows
-
on m mono-
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 :
=
-
is the weight fraction of species having a
molecular weight M;, the mean molecular weight to
be used is closer to M,, than Mv.
In fact, to our knowledge, most of the authors [33],
measured the viscosity
except references [21, 34]
molecular weight dependence using the M, mean
value. They determine the molecular weight of their
samples through the Mark Houwink equation.
The molecular weight exponent value (3.1) seems
to be closer to reptation prediction (3) than the 3.5
Bueche prediction [35], in the case of high molecular
where wi
-
-
weight (M
&#x3E;
Using the Flory equation for the expansion factor,
following reference [29], near 0 temperature, the
variation of g, with the reduced temperature r.
or
is
106).
4. Conclusion.
These sets of experiments have
demonstrated a certain number of facts concerning
the viscosity behaviour of semi-dilute solutions for
which the monomer concentration is less than 10 %
but still higher than c*.
a) The relative viscosity does not depend on the
glass transition temperature of the undiluted polymer.
b) Since the temperature variation of a 9 % PS-
benzene solution, having a ratio
of c-c* I 2IS - identical
viscosity, we may conclude that
the local viscosity is that of the solvent. This confirms
the NMR experimental results [36].
we have considered that the effective
was defined through ç
of
a
monomer
lg’
length
and that the excluded volume effect is effective at any
scale. In particular
In part 1,
=
(A1) and (A. 3)
are
compatible only if :
Substituting this expression
(2) giving ç as a function
mula
of I (A. 4) in the forof c and I we obtain :
to that of the solvent
c) The relative viscosity of semi-dilute solution
in good solvent, seems to be, in the case of PIB-toluene,
in agreement with de Gennes theory. This fact confirms
the experimental results [21]obtained a long time ago.
using
and
556
and
U jkT, k is the Boltzmann’s constant
bending energy per monomer. The temperature dependence of the quantities g/gt and q, are :
Usually m
=
The. viscosity of the solvent
the values given in reference
following relations :
was
[17].
calculated using
We obtain the
and U the
Appendix
NUMERICAL VALUES
6, 7 ; THE RELATIVE
2
CORRESPONDING TO FIGURES
VISCOSITY ACCURACY IS
2
2, 3,
%.
Appendix 3
TEMPERATURE DEPENDENCE OF THE RELATIVE VISCOSITY
OF PIB-CYCLOHEXANE SYSTEMS AT DIFFERENT CONCENTRATIONS.
References and notes
[1] BERRY, G.
C. and Fox, T. G., Adv.
Polym.
Sci. 5
(1968)
261.
[2]
DE
GENNES, P. G., Scaling concept
(Cornell University Press)
[3]
c* ~ N/R3 ,N
in
polymer physics
1979.
is the number of monomers in
a
chain of
radius R. The concentration in section 1 is expressed in volume number density of monomers, in
sections 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.
17 (1979) 1971.
[7] FLORY, P. J., Principles of Polymer Chemistry (Cornell
University Press, Ithaca N.Y.) 1953.
[8] COTTON, J. P., J. Physique Lett. 41 (1980) L-231.
[9] LE GUILLOU, J. C., ZINN-JUSTIN, J., Phys. Rev. B 21
(1980) 3976.
557
[10]
do not make a distinction between the static
v, and the effective dynamic exponent,
03BDH, because the latter plays a minor role in the
concentration exponent value.
DAOUD, M., JANNINK, G., J. Physique 37 (1976) 973.
COTTON, J. P., NIERLICH, M., BOUÉ, F., DAOUD, M.,
Here
we
exponent,
[11]
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
[14]
[15]
[16]
which the second virial coefficient is zero, for an
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Details on the apparatus will be published :
ADAM, M., DELSANTI, M., MEYER, R., PIERANSKI, P.,
[17]
J. Phys. Appl.
For each solvent we have used the viscosity temperature
dependence
cient is insensible to the exact value of effective
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[28] We know from reference [12] that the concentration
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1229.
PS-benzene and
6
Sci.
PS-cyclohexane :
DECKER, D., Thesis, Strasbourg (1968).
PIB-cyclohexane and
PIB-03B8 solvent
(IAIV) :
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[41] For numerical values corresponding to figures 2-3-6-7,
see appendix 2.