Chapter 14
Rheology of Epoxy-Amine Systems Near the Gel Point 1
2
2
Jean Pascal Eloundou , Jean François Gérard , and Jean Pierre Pascault 1
Laboratoire de Mécanique des Matériaux et Construction, Ecole Nationale Supérieure Polytechnique,B.P.8390,Yaoundé,Cameroun Laboratoire des Matériaux Macromoléculaires, Institut National des Sciences Appliquées deLyon,20 AvenueAlbertEinstein, 69 621, Villeurbanne Cédex, France
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2
The behavior of epoxy - amine systems was studied by rheology near the gel point. At the gel point, the moduli G' and G" verify the equation G'(ω)
G"(ω)
Δ
ω.
Near the gel
point, viscosity, η, and elastic modulus, G', follow power laws ε ;G ε ). of the relative distance from the gel point ( η -k
z
The exponents Δ, k and z are related one another by the percolation theory and found to be the same as those found in the Rouse model for temperatures higher than
Many authors have studied theoretically (1-5) and experimentally (6-14) the rheological behavior of physical of and chemical gels. Chemical gels, like epoxy - amine systems, formed of covalent bonds, are irreversible. In this case, polymerization produce a three - dimensional network when one monomer have a functionality greater than 2. This paper describes theory of percolation and the rheological behavior of
© 2003 American Chemical Society Bohidar et al.; Polymer Gels ACS Symposium Series; American Chemical Society: Washington, DC, 2002.
205
206 two epoxy - amine systems near the gel point. A low-T (flexible) epoxy - amine system based on diglycidyl ether of 1,4 butanediol (DGEBD) with 4,9-dioxa1,12-dodecanediamine (4D) and a high - T (rigid) epoxy - amine system based on diglycidyl ether of bisphenol A (DGEBA) with 4,4 -methylenebis [3-chloro, 2,6-diethylaniline] (MCDEA) were considered. The maximum glass transition temperature of the fully cured network, T is low for DGEBD/4D system, which presents only gelation transition at curing temperatures considered (15). T is high for DGEBAJMCDEA system, which exhibits gelation and vitrification below T (16,17), but only gelation above T . The kinetics of these systems is well known (15-17). g
g
T
gao
goo
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g00
goo
Background The percolation model (18-20) was developed to describe the properties of branched polymers near the gel point. In the bond percolation model, the monomers that occupied the sites of dimension, d, are randomly linked with the probability, x. In the case of epoxy - amine systems, x is the degree of conversion of epoxy functions. At the first step of the curing, when x is low, some dimmers and trimmers develop with chains of low molar masses. As x increases progressively, larger clusters appear (branched polymers). A low increase of viscosity, 77, and of weight average molar mass, M is then observed as amine and epoxy functions disappear. For a critical value, x i the viscosity quickly increases leading to an elastic behavior, which can be observed in rheological measurements. An infinite cluster appears, having a size and a mass that diverge at the gel point. The gelation phenomenon is characterized by the presence of two phases in the reaction medium: the soluble one (small clusters) and the non-soluble one (infinite cluster). Beyond the gel, the soluble fraction decreases while the gel fraction, n , increases. Above x very large clusters connect to the infinite one to form a network. The finite clusters with an average size decreasing as the reaction proceeds, are included in these very large clusters. W)
ge
g
geh
Structural Parameters The size distribution of the clusters (number of clusters of mass m) is N(m) oc m~ f(m/M ) T
z
where M is the z-average molar mass, which has the z
same critical behavior as the mass of the larger cluster before the gel point. f(m/M ) is a cut-off function, which implies that, a mass greater than M has a 2
z
mlM
zero probability to exist. In the mean field theory, f(m/M ) = e~ z
--. Beyond
the gel point, M , is equivalent to the gelfraction,n . The mass m is expressed as z
g
Bohidar et al.; Polymer Gels ACS Symposium Series; American Chemical Society: Washington, DC, 2002.
207 df
a function of the radius of gyration by m oc R where df is the fractal dimension of the polymer. The radius, £ of M is called the correlation length. Near the gel point, several quantities diverge as power laws of the relative distance to the gel point, s (8 = \x-x \/x i)This is the case of % ( ^ £ ~ ) , of M z
v
ge
dj
S(i
z
l/a
7
(M oc % x s~ ) and of the mass average molar mass M (M z
w
oc s~ with
w
y = (3 - T)/ x ), the equilibrium modulus, gel
z
G, varies as Gcc£ . Considering the electrical analogy rj corresponds to the conductivity of a mixture of superconductors and conductors, and x is the supraconduction threshold. G corresponds to the conduction of a mixture of conductors and insulators and x is the conduction threshold. In this way, three-dimensional simulations lead to k = 0.75 ± 0.04 (18) and z = 1.94 ± 0.01 (19). In the Rouse model approach, the macromolecule is considered as a bead spring free of any hydrodynamic interaction with the solvent, which then can be of the same nature as the polymer. This approach can be applied to the case of a get
ge!
Bohidar et al.; Polymer Gels ACS Symposium Series; American Chemical Society: Washington, DC, 2002.
208 reaction proceeding in bulk. Then, the viscosity is purely entropic and varies as 2
the mass-average square of the radius of gyration, (^R ^ (18). The calculations made in this case lead to 77 oc M ~ 2
T+2/df
2
2/d
oc < -*+ r>'°
z
^ fi-*>. j
e
e
n
three-
dimensional percolation model, k = p-2v, thus ^ = 1.33.
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df
Close to the gel point, the equation m oc R shows that the clusters forming the "polymolecular medium" are described in terms of fractal geometry. The fractal domain corresponds to scaling lengths included between the monomer size and the correlation length, £ which varies as the size of the largest cluster (20). If the fractal dimension describes the way the object occupies the volume, however, it gives no indication on the connectivity. Then, a spectral dimension, d , was introduced (1,2) which reflects connectivity and takes into account the diffusion as well as the transfer phenomena in the network. This spectral s
d,/2
dimension is given by N oc t t
where N is the number of distinct sites visited t
during a random walk in the network. The number of steps executed is proportional to the time, t, of the random walk and is connected to the average radius, R of the visited space by means of W9
w
the fractal dimension of the random walk in the network (t oc Rj* ). The three dimensions d
fy9
d and d verify the relation d = 2d / d . s
w
s
f
w
By comparing the percolating system with a mixture of conductors and insulators beyond the gel point, we obtain (1) d =2d /[2 + (z-/3)/v]. s
f
Alexander and Orbach (1) have observed, from various simulations, that the spectral dimension is always close to 4/3 in the case of percolation with a space dimension, d, between 1 and 6. For z = 1.94, the value d = 1.35 was found. The existence of a fractal domain is expressed by a distribution of relaxation times, X (time taken by a cluster to diffuse on a length equal to its own radius), from a lower limit, X , to an upper limit X (20). The upper limit eliminates the problem of the divergence of the largest cluster size when approaching the gel point, and the lower limit takes into account the similarity loss for very small dimensions. s
0
z
The viscoelastic measurements showed that, at the gel point, the distribution A
of relaxation times obeys a power law H(X) dlnXoz X' dlnX
(6-8). The
shear relaxation modulus, which describes the relaxation strain proceeding from a constant shear deformation, can be obtained by G(t)-
t/x
[[H(X)/X]e dX
with X < t < X (21). When applying the Fourier transform to this equation, we 0
z
A
obtain G(co)ozco with 1 /X < co < 1 / X (21). It appears, then, that G'and G" z
vary
according
to
a
power
0
law of
the
angular
frequency, co
Bohidar et al.; Polymer Gels ACS Symposium Series; American Chemical Society: Washington, DC, 2002.
209 A
(G'(co)ccG"(co)ccco
with co* G£&4/Jeffamine D400 system; A = 0.70 ± 0 . 0 5 , k = 1.4 ±0.2 and z = 2.8 ± 0.2 for Z)G£5^/diethanolamine system (11)). 0
gco
0
Table 3. Exponents A, k and z at Temperatures Greater than T, Downloaded by FUDAN UNIV on December 8, 2016 | http://pubs.acs.org Publication Date: October 15, 2002 | doi: 10.1021/bk-2002-0833.ch014
0
Systems DEBD/4D DGEBA/MCDEA Rouse model
k 1.44 + 0.02 1.43 + 0.02 1.33
A 0.7010.01 0.69 ±0.01 0.67
z 2.65 + 0.02 2.51 + 0.04 2.67 0
Figure 13. Exponents 4 k, z versus temperature (12, 13) 0
Temperatures Lower than T
gao
(Gelation and Vitrification)
These temperatures concern only the DGEBA/MCDEA system. From 170°C to 150°C, z decreases while A and k remain almost constant. This variation means that z , which governs the behavior of the reaction bath after the gel point, is more sensitive to the vitrification than A and k. Mean values of A and k (0.69 ±0.01 and 1.43 ±0.03 respectively) show that the reaction bath still follows the Rouse model before the gel point. 0
0
Bohidar et al.; Polymer Gels ACS Symposium Series; American Chemical Society: Washington, DC, 2002.
217 disturbed by vitrification. Motions of molecular chains become slow with relaxation times larger and larger. This behavior favors the elastic modulus G' to the detriment of the loss modulus G". The consequence is the decrease of the loss factor tand and exponent A with the temperature. Then the molecular bath is no longer a polymolecular distribution offractalsat temperatures slower than 150°C.
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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
Alexander, S.; Orbach, R. J. Phys. (Paris) Lett. 1982, 43, L625 Muthukumar, M. J. Chem. Phys. 1985, 83, 3162 Hess, W.; Vilgis, T. A.; Winter, H. H. Macromolecules, 1988, 21 2536 Rubinstein, M.; Colby, R.; Gillmor, J. R. Polym. Prepr., Am. Chem. Soc. Div. Polym. Chem. 1989, 30, 81 Martin, J. E.; Adolf, D.; Wilconxon J. P. Phys. Rev. 1989, A39, 1325 Durand, D.; Delsanti, M.; Adam, M.; Luck, T. M. Europhys. Lett. 1987, 3, 277 Chambon, F.; Winter, H. H. J. Rheol. 1987, 31, 683 Izuka, A.; Winter, H. H.; Hashimoto, T. Macromolecules. 1994, 27, 6883 Matjeka, L. Polym. Bull. 1991, 26, 109 Miaoling, L. H.; Williams, J. G. Macromolecules 1994, 27, 7423 Adolf, D.; Martin J. E. Macromolecules 1990, 23, 3700 Eloundou, J. P.; Fève, M.; Gérard, J. F.; Harran, D.; Pascault, J. P. Macromolecules 1996, 29,6907 Eloundou, J. P.; Gérard, J. F.; Harran, D.; Pascault, J. P. Macromolecules 1996, 29,6917 Eloundou, J. P.; Ayina, O.; Noah Ngamveng, J. Eur. Polym. J. 1998, 34, 1331 Eloundou, J. P.; Fève, M.; Harran, D.; Pascault, J. P. Angew. Makromol. Chem. 1995, 220, 13 Girard-Reydet, E.; Riccardi, C. C . ; Sautereau, H.; Pascault, J. P. Macromolecules 1995, 28, 7599 Eloundou, J. P.; Ayina, O; Ntede, N. G.; Gérard, J. F.; Pascault, J. P.; Boiteux, G.; Seytre, G. J. Polym. Sci. Part B: Polym. Phys. 1998, 36,2911 de Gennes, P. G. C. R. Acad. Sci. Ser. B 1978, 286, 131 Derrida, B.; Stauffer, D.; Hermann, H. J.; Vannimenus, J J . Phys. (Paris) Lett. 1983, 44, L701; 1984, 45, L913 Stauffer, D.; Coniglio, A.; Adam, M. Adv. Polym. Sci. 1982, 44, 103 Ferry, J. D. Viscoelastic properties of Polymers, 3 ed.; Wiley: New York, 1980 Hodson, D. F.; Amis, E. J. Phys. Rev. A 1990, 41, 1182 Graessley, W. W. Macromolecules 1975, 8, 186 rd
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