Macromolecules 1981,14, 1668-1676
1668
Concentration and Molecular Weight Dependence of Viscoelastic Properties in Linear and Star Polymers V. R. Raju,* E. V. Menezes,* G. Marin,* and W. W. Graessley Chemical Engineering and Materials Science Departments, Northwestern University, Evanston, Illinois 60201
L. J, Fetters
Macromolecules 1981.14:1668-1676. Downloaded from pubs.acs.org by UNIV OF WINNIPEG on 01/23/19. For personal use only.
The Institute of Polymer Science, University of Akron, Akron, Ohio 44325. Received May 7, 1981
ABSTRACT: Superposition principles have been applied to the dynamic moduli, 0'( ) and G"M, of five species of linear polymers and their concentrated solutions to establish forms for the terminal response in the limit of long chains and high concentrations. Below oim, the frequency at which the loss modulus assumes its maximum value Gm", reduced curves of G '(cv)/ Gm" and G "( )/ Gm" vs. u)/(vm settle into limiting forms rather quickly ( 5M„, where Me is the entanglement molecular weight). Above ), for several linear species at various molecular weights and concentrations. Superposition of dynamic moduli for star-branched polymers at different concentrations is also examined.
The linear viscoelastic properties of narrow-distribution polymers at high concentrations have been the subject of numerous studies.1,2 Separation of the relaxation spectrum H{t) into distinct terminal and transition regions for linear chain systems is well established. The separation first appears when the molecular weight M exceeds Mc, the characteristic molecular weight for zero shear viscosity 0 at that concentration, and then increases rapidly with increasing chain length. Moreover, the behavior of the plateau modulus GN° and the steady-state recoverable compliance Je° suggests a remarkable similarity in the form of the terminal spectrum H^r) for different polymer » Mc, these quantities and the viscosity species. For can be expressed as moments of the terminal relaxation spectrum alone:2 GN°
Vo
Je°
=
=
=
X>M d In
(1)
d In
(2)
X^TffiM ~2
%( ) d In
The product GN°Je° is therefore of the terminal spectrum
GM
=
XVffiÍT) d In /
^
a measure
^
Experimental Samples and Procedures Six structural species prepared by anionic polymerization are represented: linear polybutadiene (designated by L),3,4 fully hydrogenated linear polybutadiene (designated by HPB or PHPB),6 polybutadiene stars (designated by S),3 linear polystyrene (designated by C),6 linear polyisoprene (designated by PI),7 and fully hydrogenated polyisoprene (designated by HPI).7 Hydrogenated linear polybutadiene has the polyethylene structure except that it contains ~18 ethyl branches/1000 backbone carbon atoms. It is virtually indistinguishable from linear polyethylene in the melt state.5 Hydrogenated polyisoprene is equivalent to a strictly alternating copolymer of ethylene and propylene although it contains about 20 isopropyl side branches/1000 backbone carbon atoms/7 All samples have narrow molecular weight distributions (Mw/Mn < 1.1). Molecular characterization data are given in Table I. The diluents used for polybutadiene are listed in Table II. The first four are commercial hydrocarbon oils. Tetradecane is a good solvent for polybutadiene, isobutyl acetate is a solvent ( = 20 °C), and chain dimensions with the low molecular weight polybutadiene as solvent should be near the unperturbed values. Glass temperatures for the solvents range from well below to well above Tt = -95 °C for high molecular weight polybutadiene. Two low molecular weight linear polyethylenes with narrow distributions (Bareco waxes, Petrolite Corp.) were used as diluents for the hydrogenated polybutadienes (see Table I). The polybutadiene solutions were prepared by dissolving the diluent and polymer in an excess of benzene and then stripping out the benzene under vacuum at 25 °C.3 The hydrogenated
(3)
of the breadth
) d In
) d In r/fHM d In
-
n
(4)
and practically the same value, GN°Je° = 3.0 ± 0.5, is found for different polymer species. The product is independent Present address: Bell Laboratories, Murray Hill, NJ 07974. Present address: Ethicon, Inc., Somerville, NJ 08876. * Present address: Laboratoire de Thermodynamique, Université de Pau, F-64000 Pau, France. *
4
0024-9297/81/2214-1668$01.25/0
©
1981 American Chemical Society
Vol. 14, No. 6, November-December 1981
Viscoelastic Properties in Polymers 1669 The hydrogenated polybutadiene systems did not reach the limiting storage modulus behavior,5 so Je° could not be obtained in those cases. The results are given in Tables III-V. Solvent viscosities were measured with calibrated Cannon-Ubbelohde capillary viscometers (Table II).
Table I Molecular Characteristics of Polymers Included in This Study
[ Ithf-
AMJ
(Mw )gpc X 10'4
mAgpc
0.46 20.0 23.0 34.0 35.0 51.7 81.3
dL/g 2.76 4.02 2.41 2.92
1.0 2.0
bHx*(r/a)
=
where Hx* is the terminal spectrum in some convenient reference state and a and b are shift factors (a = b = 1 in the reference state) which depend on polymer and diluent species as well as T, M, and . Equations 1-3 yield the following relations:10
0.99 1.27
MIMe
must be expressable
cases
as
4.63,
(5)
(6)
=
0 at
b(Gm")*
a>m],
(13) (14)
(tom)*/o
(Gm"/GN0)* and wem/GN°
=
=
(VouJ
Gn0)* must also be universal constants if eq 7 is valid. The quantities GN°, Gm", 1 /Je°, and umn0 are proportional to
the modulus shift factor b alone and so must vary with , M, and T in the same manner. Finally, from eq 11-14, the dynamic moduli plotted in reduced variables, G'/Gm" and G"/Gm", vs. cu/ 1 are less strongly dependent on dilution (decreasing M/Me; see below) than in the other species. There are also some experimental difficulties in obtaining accurate values of G" at high frequencies in undiluted HPB, owing to its unusually large plateau modulus.9 Until such matters are cleared up, we prefer to omit the HPB data altogether in the establishment of limiting forms. The loss modulus for a single relaxation time process (single Maxwell element)
G"
Table V Rheological Properties of Solutions of Star-Branched Polybutadienes at 25 °C sample 0, P =/e°, cm2/dyn S200in F391 1.000 9.50 X 10’ 1.29 X 10"6 0.812 1.38 X 107 1.50 X 10"6 1.45 X 106 0.609 2.07 X 10"6 0.403 1.15 X 10= 3.30 X 10"6 0.206 8.62 X 103 6.54 X 10'6 0.108 1.45 X 10"5 1.04 X 103 1.38 X 10= 0.0526 S100 in F391
S200 in TD
1.000 0.763 0.538 0.409 0.197 0.105 0.0533
8.98 2.14 4.54 1.66 8.61 1.52 9.90
0.656 0.405
4.02 6.66
X
10s
X 105 X 104 X 104
102 X 102 X X
10‘
X
10s 103
X
1.21 1.47 1.76 2.50 5.38
X
2.05 3.45
X
X
X X X
X
10"6 10"6 10"6 10"6 10"6
10"6 10"6
,.0. Values of to be good even for values of / entanglement molecular weight were calculated for each species with1
Me
for 0 = sample.
/G^
=
Values of M/Mt are given in Table I for each (If M/Mc were used, all values would be lower by
factor of approximately 2.2) The smooth progression of the curves with increasing M/Me indicates a decrease in a
sample
L200 L350 HPB-3500
PI-lb
HPI-1c C6bbd C7bbd
n0, P
5.41 3.35 1.12 5.75 2.97 3.10 1.50
X X
X X
X X X
106 107 107 10s 107 10s 10s
1
+
( / )2
especially at high frequencies. There are some processes with relaxation times greater than l/wm and apparently a tail of shorter relaxation time processes extending even below 0.1/ Mc), presumably independent of chain length (for should decrease much more rapidly than this. The results strongly suggest that intermediate processes are involved, such as, for example, the equilibration step of the DoiEdwards model.11 Such processes might shift with chain length, but less rapidly than true terminal processes, and might therefore be more accurately described as plateau relaxations. The area under the limiting curve in Figure 1 gives, with ~
eq 17
Gn°
=
(20)
3.56Gm"
This result is somewhat different from the Marvin-Oser equation, GN° = 4.83Gm", which is based on a shifted Rouse model for the terminal spectrum.1 Values of GN° calculated with eq 20 are given in Table VI for the various species.
Table VI Rheological Properties of Undiluted Polymers Gn°, dyn/cm2 Je°, cm2/dyn 1.80 X 10"7 1.15 X 107 2.0 1.18 X 107 3.0 1.92 X 10"7 2.2 2.21 X 107 5.6 6.30 X 10"7 3.26 X 106 3.6 1.13 X 107 1.99 X 10"7 5.6 1.20 X 10"6 1.78 X 106 1.25 X 10"6 1.78 X 106 1.1 b
(19)
_
is shown by the dashed line in Figure 1. Although the limiting behavior (solid curve) corresponds to a narrow spectrum, it is still broader than a single-line spectrum,
(18)
1.
_
Gm”
~
•VGN°
s"1 X
10°
X
10"1
X X
10° 10° 10'1
X
10"'
X
10"2
X
0.94 0.85 1.11 0.99 0.95 0.98 0.93
2.07 2.27 2.05 2.25 2.14 2.23
HPB data shown were collected at 130 °C. Polyisoprene data at 25 °C were obtained from ref 7. c Hydrogenated polyisoprene data at 25 °C were obtained from ref 7. d Polystyrene data at 170 °C were obtained from ref 6. 0
Figure 4. Reduced storage moduli for several narrow-distribution Figure 2. Reduced loss moduli curves for polybutadiene solutions (sample L350 in F391) at 25 °C. The symbols denote volume fraction of polymer: (p) 1.000; (0)0.432; (0) 0.222; (0)0.107; (O) 0.0854; (O) 0.0578. The dashed line represents the limiting behavior obtained for undiluted high molecular weight linear polymers in Figure 1. The values of Af/Me shown were calculated
with
eq 18.
3. Reduced loss moduli curves for hydrogenated polybutadiene solutions (sample HPB-350 in Bareco 1000) at 130 °C. The symbols denote volume fraction of polymer: (Ó) 1.000; (O) 0.828; (0) 0.662; (0) 0.498; (0)0.310; (0)0.211. The dashed line has the same meaning as in Figure 2.
Figure
are small except for polystyrene, which is reduced about 12%, and HPB, which is increased about 5%. Equation 20, of course, only applies to data on linear polymers with very narrow molecular > Mc. weight distributions and Figures 2 and 3 show similar reduced plots for solutions of polybutadiene and hydrogenated polybutadiene. The pattern is similar to that in Figure 1. Superposition is good for
