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Viscosity and longest relaxation time of semi-dilute polymer 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. Good solvent. Journal de Physique, 1983, 44 (10), pp.1185-1193. �10.1051/jphys:0198300440100118500�.
�jpa-00209702�
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 TR
sont effectuées dans des solutions semi-diluées de polystyrène-benzène. Lorsque la concentration c est inférieure à 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 du premier 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 benzene
solutions. For monomer concentrations c smaller than 10% and over a large range of the reduced concentration
c/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 time
T1 of a single chain are function of the reduced concentration c/c* only;
2014 the shear elastic modulus is independent of the molecular weight
These 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 explained using the simple reptation model.
Classification Physics Abstracts
46.30J - 52.40 - 61.40K - 82.90
1. Introduction.
De Gennes’ theory of the viscosity of a semi-dilute
polymer 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 overlap
concentration 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 function of the reduced variable c/c* only. If clc* > 1, the relative 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 viscosity predicted by reptation theory [3] (XM = 3), imposes
the Xc exponent value :
Thus :
The longest viscoelastic relaxation time of the
polymeric system TR, called the reptation time, is the
time needed for a chain to reptate the length of a
fictitious tube. It represents the hindrance to the chain 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 the longest relaxation time TR :
where G is the shear elastic modulus.
2. Experimental conditions.
2.1 SAMPLE PREPARATION. - The solvent used is benzene of analytical grade (RP) and the polymer is polystyrene.
The monomer concentration [4] of a semi dilute solution lies between the overlap concentration c*
and an upper concentration S When the monomer
concentration exceeds i (e - 10 % for polystyrene benzene) the solutions no longer follow the laws
obtained for the semi-dilute case. Actually, the polymer
mutual diffusion coefficient is an increasing function
of concentration [5] in the polystyrene ethylbenzene system up to 20 %, where it becomes stationary before decreasing. The correlation function of the light
scattered by a polystyrene benzene solution has an exponential line shape only if c 10 %. In addition
the benzene mutual diffusion coefficient [6] has an
activation energy independent of concentration, only
if c 10 %.
Our usual definition [7] of the overlap concentration
c*, for polystyrene benzene solutions, is :
our definition of c* corresponds to the concentration at 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 are
given in table I. Some measurements were also done on
low molecular weight polystyrene, at high concen-
tration (see Table I).
Samples are prepared in the measuring cell a long
time in advance (at least two months). No mechanical
force is exerted on the polymer in order to facilitate
the dissolution; we check that there is no concen-
tration gradient by measuring the viscosity at various points of the cell.
Table I. - Molecular weight, polydispersity index and overlap concentration c* of the samples studied. Poly- styrenefurnish by : (-) Toyo Soda inc. (Japan), (*) CRM Strasbourg (France).
2. 2 APPARATUS. - In order to measure the zero shear
viscosity and the longest relaxation time we use a
magnetorheometer described in reference 9. Here
we only recall the main features. A magnetic sphere
immersed in the sample experiences a magnetic force provided by a current passing through a coil. The
current 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 a speed v, the measurement of the current intensity I passing through the coil allows us to determine the viscous force :
where Io is the current intensity needed to counter-
balance the gravitational force and r is the radius of
the 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 our
We obtain t e absolute value of the viscosity of our samples by a calibration of our apparatus with stan- dard silicon oil (’1 = 40.47 poises at 35 OC).
A range of viscosities, between 0.1 and 101 poises,
are measurable within a precision of 2 %. The
relative viscosity q, is calculated using for the benzene viscosity 110 :
obtained from tabulated values [10].
This apparatus allows us also to measure the longest relaxation time of the polymeric system. After the motor is stopped, the variation of the current
intensity 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 the
moment 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 a
least 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 and
50 ms. The lower limit is imposed by the present characteristic coefficient of the electronics and by the amplitude of the signal which, being proportional to
the macroscopic viscosity, becomes too small to be
detected when the viscosity is smaller than 2 poises.
Fig. 1. - Variation of the current intensity passing through
the coil in arbitrary units as a function of time. The full line is the best fit corresponding to a characteristic time of 0.184 s (the sample is : Mw = 6.77 x 106, c = 4.2 % at
35 oC).
The experimental treatment, using a single expo- nential decay for the current intensity, implies that
the longest relaxation time is much larger than all the others 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 amplitudes
vary as l/p2. Therefore, experimentally the system is sensitive to the first mode only since the second mode has an amplitude and a characteristic time which is 10 times smaller than the first one.
In the semi-dilute polymeric system there is also a spectrum 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 our
experimental conditions (c > 4 c*) the reptation time TR is much larger than Tj.
The experimental results obtained, concerning the
shear elastic modulus (see section 3 .1. 3) allow us to be
confident that we are measuring the longest relaxation
time with :
The sample is thermalized by water circulation;
the homogeneity and the stability of the temperature is 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 relaxation
time (5 x 10- 2 s TR 100 s).
3. Experimental results.
Two sets of experiment were carried out in order to
test :
- the concentration dependence of ", T R and G.
Using several molecular weights, the monomer con-
centration was increased from c* to 10 %. For the
two lowest molecular weights, the experiments were performed up to 20 %.
- the molecular weight dependence of q and TR.
At 3 given concentrations (2, 5 and 8 %) the molecular
weight was increased from 1.26 x 106 to 20.6 x 106.
In sections 3.1 and 3.2 we give the experimental
results obtained in a good solvent (benzene) for ", TR, and G. They are discussed in section 3 . 3. The raw
results are given in the appendix.
3.1 CONCENTRATION DEPENDENCE.
3.1.1 Relative viscosity. - Whatever the molecular
weight, in the range 1.7 x 101 and 20.6 x 106, the
relative 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 reduced concentration (log-log scale). The following symbols are
used 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 molecular
weight 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, could possibly be only qualitative because discrepancies
can be hidden by the large variations of the observed
viscosity. To get rid of this difficulty we use another representation : 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 of
c/c*, we observe that for the highest molecular weights (M > 106), the points obtained with different mole- cular weights do not deviate systematically from each
other.
Fig. 3. - Variation of the measured quantity tl, divided by
% calculated using equation 9. Symbols are the same as in figure 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 points allows, continuously. Figure 4 gives the variation
of X(c/c*) calculated for each molecular weight as a
function of c/c*. The full line corresponds to the
results 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 the results obtained at c/c* > 10, leads to a stable exponent value equal to 4.46 ± 0.05 in agreement with previous
results [12]. In figure 4, the points corresponding to the
results obtained with the two lowest molecular weights
at a concentration lying between 8 and 20 % are systematically above the full line.
We have also considered the variation of X(clc*),
obtained at given values of c/c*, as a function of concentration. The variation of Xe with concentration
gives rise to a family of curve each one corresponding
to a value of c/c*. Each curve is normalized by its
value 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 normalized
quantity X, .IX,*,,-o no longer has a constant value
Fig. 4. - Variation of concentration exponent X c as a function of c/c*, the full line represents Xc obtained with
highest molecular weights, the arrow indicates the theoretical value. Symbols are the same as in figure 2.
Fig. 5. - Variation as a function of concentration (in g/g)
of the effective exponent Xc divided by its value X,,.,c-O
obtained from the semi-dilute regime and extrapolated to
zero concentration. Symbols represent different values of
c/c* : c/c* = 18., 7.38 0, 5 x, 3.5 EL 2.5 p.
(Fig. 5). This concentration corresponds to the upper concentration c.
These observations will be discussed in sec- tion 3. 1. 3.
3.1.2 Longest relaxation time. - Due to the lower limit of the relaxation time TR measurable with this apparatus (50 ms), the measurements can only be
done on samples having a reduced concentration greater than 7, thus on a smaller range of concentration than in the preceding section. In all cases, the current intensity in the magnetic coil relaxes to equilibrium exponentially (see Fig, 1).
As an example, in figure 6, we give, the variation of
TR as a function of concentration, obtained with the molecular weight 20.6 x 106. Typically a sample
whose concentration is 2.66 % has a longest relaxation
time 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 using Mw = 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 (obtained
with v = 0.588) equal to 1.6.
We calculate T 1 using the relation :
where kB, T and A are the Boltzmann constant, the absolute temperature and Avogadro’s number, respec-
tively ; a, a numerical constant which is taken to be equal to 0.42 as in the case of a gaussian chain with hydrodynamic interactions [13]. [tl], is the intrinsic
viscosity, which, in the case of polystyrene benzene
solution [14], is :
The ratio TRI T, is plotted as a function of the
reduced concentration c/c* (Fig. 7), yielding a single
curve independent of the molecular weight and dependent on c/c*, only. The fit to the experimental
data gives :
The important experimental result obtained from
figure 7 is that the reduced concentration c/c* is a
reduced variable for the ratio TRIT,, in agreement with de Gennes’ theory (Eq. 6). However the experi-
mental value of Yc = 2.05 is much higher than the
theoretical 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 exponent y c = 2.05 ± 0.1 (see Eq. 11). Symbols are the same as in figure 2.
1190
3.1.3 Elastic shear modulus. - From our experi-
mental determinations of the viscosity, the longest
relaxation time, and the relation 7 : G = j7/ TRI we
deduce the shear elastic modulus.
Our determination is in good agreement with
experimental results obtained with a relaxometer of the cone and plate type [ 15] on a polystyrene aroclor
solution (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 7r have the same concentration dependence.
From osmotic pressure measurements [8], on poly- a-methylstyrene dissolved in toluene it is found that
7t - C2.32 and that at a concentration of 7.8 g/cm3 corresponding to a reduced concentration c/c* of 18.2 :
At the same concentration the shear modulus value is 2 x 103 dynes/cm2. Thus these two moduli differ
by 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 dependent
We found that :
with
Thus the shear elastic modulus and the osmotic bulk modulus 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 a
function of concentration expressed in g/cm3 [16]. The slope corresponds to the exponent Zc = 2.36 (see Eq. 13). Symbols
are the same as in figure 2, the ® is the value of the shear modulus extracted from reference 15.
respectively), they differ only by a large numerical
factor.
It has been shown [1] that in a semi-dilute solution the 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 good agreement with v values determined previously [17].
We note that the relation 7 :
implies that the concentration exponents of il, G and TR are related by : X, = Y,, + Z,. This is only
found to be true if each exponent is determined over
the same range of concentration.
Actually for c/c* > 7 :
3.2 MOLECULAR WEIGHT DEPENDENCE OF THE VISCO- SITY AND LONGEST RELAXATION TIME. - The main result 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 the
solvent
This theoretical result is not obeyed in a polymer
melt where it is well lnown that the exponent value is around 3.4 [19, 20].
In order to measure the XM exponent in semi-dilute solutions we prepared samples with given concen-
trations (2, 5 and 8 %) using five different molecular
weights (M > 1.26 x 106). At a given concentration the variation of the viscosity and of the longest
relaxation time as a function of molecular weight
allows us to determine XM.
Figure 9 shows the experimental results obtained
from samples with a monomer concentration of 8 % ;
one observes that the exponent XM is well defined and
its 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, larger
than c*, and whatever the temperature, both XM
determinations (via il, or TR) lead to the same expo-
nent value. Obviously, this corresponds to the result already mentioned above : namely that the shear
elastic modulus (G = 17/T R) is independent of the
molecular weight
At a given temperature the exponent XM is concen-
tration dependent This can be observed in figure 10
where the XM values are plotted as a function of
