European Polymer Journal, Vol. 12, pp. 847 to 848. Pergamon Press 1976. Printed in Great Britain•
TESTING OF A MATHEMATICAL MODEL FOR VISCOELASTIC BEHAVIOUR OF POLY-c~-OLEFINS U S I N G POLYISOBUTENE JEAN-YVEs DECROIX, ALAIN PILOZ et JEAN-FRANCrOISMAY Laboratoire de Chimie Macromolbculaire, Universit6 Claude Bernard, Lyon I, 43, Boulevard d u l l Novembre 1918, 69621 Villeurbanne, France
(Received 9 March 1976) Abstract--The analytical model proposed for poly-~-olefinshas been tested on polyisobutene. Relaxation modulus E(t) and distribution of relaxation times have been predicted from dynamic experiments and agree with literature data.
Table 1. Results for polyisobutene
It has been shown [1] that the complex relaxation modulus E* (i~) of a polyolefin could be analytically described in the glass-rubber transition by an equation with four constants and one variable:
E~ - Eo E*(ko) = Eo + 1 + ~(iooz) - k q- (iogz) h
(1)
where E0 is a static modulus, E~ is a dynamic modulus, k and h are parameters which describe the low and high frequency behaviours of the polymer, z--relaxation time, o)--pulsation of frequency, 6-~zonstant. From this equation the real relaxation modulus E(t) and the distribution of relaxation times H(z) can be derived through some approximation or inversion formulae. In order to test the validity of this mathematical model, we have studied the mechanical properties of the well known polyisobutene, predicted E(t) and H(r) and compared them with the many numerical data available in literature I-2-7].
k
h
0.61
0.13
Eo E~ dynes, cm- 2 dynes,era- 2 6 5
× l0 6
4.5 × 10t°
8
Correlation coefficient 0.99
0.97
where ro is the relaxation time at temperature To. The theoretical values of E(t) have been calculated at 298K by taking ro = 10-84sec (value reported by Ferry et al. [2,3]). It has then been compared (Fig. 1) with the master relaxation curve obtained from relaxation experiments by Catsiff and Tobolsky [4-6]. The predicted values of E(t) are in very good agreement with the experimental values. The biparabolic model has also been used to obtain the relaxation
log E(t)
dynes . cm -2
N
1. EXPERIMENTALRESULTS
10
Mechanical properties of a polyisobutene sample of a weight-average molecular weight 500,000 have been investigated in the fl relaxation range by using a viscoelastometer Rheovibron (model DDV II) at four frequencies (3.5, 11, 35 and ll0Hz). The Cole-Cole plot shows the typical asymetrical shape observed for other polyolefins like poly-l-butene and poly-l-pentene. So polyisobutene can be closely described by an analytical equation like (•); k and h have been calculated by using linear regression and their values are reported with those of E0 and E~ in Table 1.
x
9
8
7 11. C A L C U L A T I O N O F E(t) A N D H(z)
The real relaxation modulus has been obtained by using Ter Haar's inversion formulae [8]: -18
E(t) = (E*(p))p= 1/r E(t) = Eo +
I -16
t -14
l -12 - 1 0
Fig. 1. Relaxation modulus
E~ - Eo 1 + 6(t/Zo) k + (t/Zo~'
[ -8
I_ -6
Io9 t lhours) ~\ \ I I i~ -4 -2 0
~
E(t) of po]yisobutene at 298 K,
theoretical value (obtained from mathematical model); x - - -x data of Tobolsky [6]. 847
848
J.-Y. DECROIX,A. Pn~oz et J.-F. MAY predicted by Zimm [10]. A similar result has been obtained on another amorphous polyolefin (polyhexene-1) for which the parameters are nearly the same.
101
log H (z)
IlL CONCLUSION 9
,,~
1
The results obtained for polyisobutene confirm the validity of the mathematical description proposed and of the inversion formulae used. It is therefore possible to obtain quickly, with good accuracy, information on the relaxation behaviour of polyolefins with a single dynamic experiment.
8 7
°t 54
I
I -lO -8
-6
L -4
I -2
[
REFERENCES Io9 't (s)
le
o
Fig. 2. Relaxation spectra of polyisobutene at 298 K, x . . . . x data data of Ferry et al. [3]; - data predicted from model. spectra [1]; the slope is: k at long time (z >> %) and h at short times (z '~ %). The theoretical values fit very well with data reported by Ferry et al. [3] (Fig. 2). It is to be noted that the slope of the relaxation spectra at long time shows a significant deviation from the theoretical value predicted from the theory of Rouse [9] but it is very near the theoretical slope
1. J. Y. Decroix, A. Piloz, A. Douillard, J. F. May and G. Vallet, Europ. Polym. J. 11, 625 (1975). 2. E. R. Fitzgerald, L. D. Grandine and J. D. Ferry, J. appl. Phys. 24, 5, 650 (1953). 3. J. D. Ferry, L. D. Grandine and E. R. Fitzgerald, J. appl. Phys. 24, 7, 911 (1953). 4. E. Catsiff and A. V. Tobolsky, J. Colloid. Sci. 10, 375 (1955). 5. A. V. Tobolsky and E. Catsiff, J. Polym. Sci. 19, 111 (1956). 6. A. V. Tobolsky, Properties and Structure of Polymers, p. 157, John Wiley, New York (1967). 7. W. Philippof, J. appl. Phys. 24, 6, 685 (1953). 8. D. Ter Haar. J. Polym. Sci. 6247 (1951). 9. P. E. Rouse, J. chem. Phys. 21, 1272 (1953). 10. B. H. Zimm, J. chem. Phys. 24, 269 (1956).
R~mn~--Le mod61e biparabolique utilis6 pour les poly-~-ol6fines a 6t6 appliqu6 au polyisobut6ne. Le module de relaxation et le spectre de relaxation ont 6t6 calcul6s ~t partir de mesures dynamiques et sont en accord avec les donn6es de la litt6rature.