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Critical behavior of anhydride cured epoxies
V. Trappe, W. Richtering, W. Burchard
To cite this version:
V. Trappe, W. Richtering, W. Burchard. Critical behavior of anhydride cured epoxies. Journal de Physique II, EDP Sciences, 1992, 2 (7), pp.1453-1463. �10.1051/jp2:1992212�. �jpa-00247742�
Classjfjcatjon
Physics Abstracts 80.00
Critical behavior of anhydride cured
epoxies
V. Trappe, W. Richtering and W. Burchard
Institute of Macromolecular Chemistry, University of Freiburg, 78000 Freiburg, Germany (Received J7 February J992, accepted 8 April J992)
Abstract. Critical behavior was studied with a crosslinking system obtained by living anionic
polymerization, where the primary chain length was kept constant and the crosslinking density
was varied. Gelation was found at a critical ratio of crosslinker per chain X~ = 0.884 ± 0.004.
Different samples from the pre gel region were studied by dynamic and static light scattering in dilute solution and oscillatory rheology in melt. The exponents y=1.75 ±0.38 and
v =
0.98 ± 0.19, determined from M~ and R~ dependence on (X~ X ), are in accordance with three dimensional percolation theory. The distribution of diffusion coefficients obtained by
inverse Laplace transformation of the time correlation function shows power law behavior in a limited interval, from which an exponent v = 2,17 ± 0.03 is derived. Rheological measurements show a systematic change of G'(w ) and G"(w from typical liquid to the critical gel behavior, where tan 8
= G" (w )/G'(w ) becomes frequency independent.
1. Introduction.
In synthesis of networks two principles can be followed (I) connection of linear polymer
chains by crosslinkers (vulcanisation) or (it) polymerization of f-functional monomers with f ~ 2 (polycondensation). The process of branching may be described in detail by either the Flory-Stockmayer (FS) [1, 2] theory or by the lattice percolation model [3, 4]. The forrner is
equivalent to a mean-field approximation and allows a rather good prediction of the point of
gelation as well as conformational properties [5, 6] and post gel quantities [7]. The percolation model, often based on Monte Carlo simulation, describes the structure forrnation of branched systems by randomly distributed bonds on a lattice and is a non-mean-field approach. It allows a better prediction of the critical behavior, than the FS-theory but a description of physically meaningful absolute values is not possible since the prefactors cannot be calculated
universally. For step reactions with equal reactivity of the groups or randomly distributed crosslinkers along chains, both concepts are easily applied. In this paper we present a system
which differs from the former ones by a living anionic growth mechanism. Here the
crosslinkers are introduced successively and the question arises, if there is a dependence of the crosslinking efficiency on the already forrned structure. Furtherrnore, it is of interest
whether on such conditions the predicted critical behavior by percolation still holds true.
2. System.
The mechanism of the anhydride curing of epoxies was studied by Matejka et al. [8]
(Scheme I) and found to be an anionic living chain reaction. The resulting typical conditions, (I) DP
= a lAfo]/~lo] (it) Poisson-distribution (iii) first order kinetics were examined with the
linear system phenyl glycidyl ether/phtalic acid anhydride (PGE/PA) using I-methyl
imidazole ( I-MI) as initiator [9, 10]. The branched system described in this paper is based on the idea to start linear chains including a few branching monomers (bisphenol-A-diglycidyl
ether (BADGE)), such that the branched product can be described by DP
= DPO I/(I -X). (I)
CO
R~ ~O
R'~N + R"CH~-
~~'CH~
- R'~~ CH~ CH O~
'~~
»
CH~R"
~O
e ~ R"CH2~ CH 'CH2
R'~N CH~ CH OOC R COO .
CH~R"
e ~
R'~N CH~ CH OOC R COO CH~ CH O
CH~R" CH~R"
Scheme I. Anhydride curing of epoxies initiated by tertiary amines.
The ratio X
=
IIIADGE II ~lo] represents the number of branching points per chain and was varied while the primary chain length was kept constant at DPO
= ~lvloY~lo] = 50 ~lvlo] total concentration of epoxy monomers and ~lo] concentration of initiator). The experimental
results are shown in figure I. Equation (I) gives a simplified description of the system, based
on the idea of crosslinked, monodisperse linear chains, which can be directly related to the vulcanisation model. For polydisperse primary chains Flory [11] derived the relationship
X~ = X~ (DP~o/DP~O) (2)
where the subscripts relate to weight and number averages, respectively. Gelation will take place at X~~ = I. By experiment, however, only the number average can be asserted. We
found X~~ =
0.884, which implies a polydispersity index of DP~o/DP~O = 1,13 for the primary
chain length. This value is in very good agreement with the polydispersity found for the linear
iO~
i~
" iO' /
~j /
E
~/
~ ~~o /~
IL
' ~ /6
~ ~,'
~ iO' ,1'
O-O O-i O.2 O.3 O.4 O.5 O.6 O.7 O-B O-Q 1.O i-1
x
Fig. I. Change of molar mass as a function of the molar ratio of BADGE to initiator X
= IBADGE ]/Jo]. Solid line : theory according to equation (1).
PGE/PA-system, where the deviation from Poisson distribution is caused by an initiation rate smaller than propagation rate [9].
3. Experimental part.
The starting materials were purified and carefully dried. The polymerisation was carried out in bulk in argon atmosphere at 120-150 °C. The obtained resins were purified by precipitation
in methanol.
The refractive index increments were measured in tetrahydrofurane (THF) at 25 °C with a
Brice Phoenix differential refractometer (A
=
496.5 nm, dn/dc
= 0,1607 ml/g ;
A
=
647, I nm, dn/dc
=
0.1512 ml/g).
The static and dynamic light scattering were measured simultaneously in THF at 25 °C with
an automatic goniometer and an ALV-3000 structurator/correlator from ALV-Langen, FRG.
As light sources an argon (A
= 496.5 nm) or a krypton (A
= 647, I nm) ion laser, model 2020, from Spectra Physics were used. The data were treated on-line with the program ODIL from Eisele. Details on the instrument and data evaluation are given in reference [12].
A Rheometrics Dynamic Spectrometer (RDS model 7700) with parallel plate geometry was
used for dynamic mechanical measurements. The temperature range was 90 °C to 130 °C. The
glass transition occurs around 60 °C and depends on the degree of crosslinking.
4. Determination of X~.
Divergence of different quantities is a characteristics of the gel point (X = X~). Power law behavior is expected [3]
Y [X~ X ~ (3)
with Y being M~, R~ = (S~l'~~, 7~, E or G. It is well known, that the different exponents (z) are highly correlated to the position of X~. Thus, a precise deterrnination of X~ is strived for.
One can proceed in the following way :
a first estimation of X~ can be obtained by plotting for instance I/M~ versus
X and extrapolating I/M~ to zero ;
the exponent z can be obtained by linear regression of the data in a double logarithmic
plot of I/Y versus (X~ Xi ;
varying X~ around the first estimation and calculating the corresponding exponents, a
dependence of z on X~ is found with an error of the regression line which again varies with
X~ and passes through a minimum ;
that pair of values (z, X~), which corresponds to this minimum of the error can be considered as optimum.
For the treatment of our data we used relationship (3) with Y= M~, z= y and
Y
=
R
~, z = v.
We performed the described procedure for M~ and R~ by (I) using all data and (it) the data-set without the highest X-data-point, since this is involved with the highest experimental error. The significance level of the straight regression line was chosen to be 95 fb.
One of the obtained error curve is shown in figure 2 as a representative example. The error passes in all curves through a minimum which corresponds on average to X~
=
0.884 ± 0.004.
Sol-gel transitions can also be observed in small amplitude oscillatory shear experiments. In the pre-gel region the material behaves as a viscoelastic liquid, in the post-gel region it exhibits properties of a viscoelastic solid [13]. At the gel point the relaxation modulus follows
a power law [14, 15].
G (t) t- n (4)
O.20 O.92
o,ia o-go
~ ~ X
- n
b
o,16 oBB
O,14 O.86
2 3
0/ (Xc)
Fig. 2. Error of linear regression («) versus y with varying X~.
Accordingly the frequency dependence of storage (G'(w)) and loss (G"(w)) moduli are given by power laws with the same exponent n [14, 15].
G'(w)~G"(w)~ w~ (5)
Consequently, the loss angle 3 becomes independent of frequency at the gel point and is related to the exponent n as follows.
3
= nor/2 or tan 3
= G" (w )/G'(w ) = tan (nor/2). (6)
Near the gel point the frequency dependence of G'(w and G" (w can in a limited region also be approximated by power laws. The exponents for G'(w and G"(w ) now differ from each other and depend on the distance from the gel point (Figs. 3, 4). This provides another
ioa ~~~
/f lo?
~
io~
io~
_ ~~~ ~o
3 c
Z - ~o
~ io4 ~
IQO
3 lO~
~~
lo ~
lo' io'~
io~io~~io~~io~'io° lo' io~ io~ io~ io~ io~
a~ cu / rad s~~
Fig. 3. Mastercurves of G'(O), G"(+) and tan (6) at X
=
0.80.
io~ io~
~~7
~ CL
~
io~
lo'
_ ~~~ ~o
~
- i
~5 lo' ~
io°
3 io~
b
io~
~~i ~~-i
io~'io~~io~~io~'io° io~ io~ io~ lo' io~ io~
a~ cu / rad s~~
Fig. 4. Mastercurves of G'(O), G"(+) and tan &(6) at X
=
0.87.
method for the X~-deterrrination. The different exponents can be plotted versus X, and a
point of intersection can be found by extrapolation. In the present case (Fig. 5) this occurs at
X~ = 0.89 ± 0.01 and n
=
0.71 ± 0.01.
This technique is, of course, not as accurate as the described minimization, but the obtained value is close to X~
=
0.884 and confirms the position of the gel point.
2
o
~g o
2 ~
l + + q~,
~
+
~ o,
~ +,+ g
2 ~
o
o.oo o.50 1-cc
x Fig. 5. Exponents for G'(O) and G"(+) versus X.
S. f~ritical behavior.
According to three dimensional percolation and mean field theories the following exponents
are predicted [3]
Mean field Percolation
(melt) (swollen)
y =
1.0 y =1.80 y
= 1.80
v = 0.5 v
= 0.88 v
=
I-I I
v/y = 0.5 v/y = 0.49 v/y
=
0.62
Using the evaluated X~
= 0.884, we find for the present system : y = 1.75 ± 0.38
v = 0.98 ± 0,19.
These exponents are close to the percolation prediction, although the fairly large error is still
unsatisfactory (see Figs. 6a, fib). The main error clearly results from the uncertainty of X and thus of the critical region (X~ X ), which can be eliminated, resulting in
R~ M('Y MS (7)
io~
~
~
10~
~ a ,"'
,'
~ ,'
,' ,' ,"
,"
io~7 ,'
io~
io~~ io~' io°
xc x
a)
lO~'
h ,,"
w ,'
iQ~2 ,,"'
,' ,,"
,' ,"
~~-3
io~~ io~' io°
xc x
b)
Fig. 6. Plots of the reciprocal weight averaged molar mass and radius of gyration versus the distance from the gelation threshold. Solid lines : linear regression, the dashed lines indicate the slope according
to mean field theory.
Direct measurement ofR~ as a function of M~ as to see in figure 7 shows a power law with an exponent b =
0.56 ±0.03. The error found by this method is much lower than that one
obtained by the experimental values for y and v, b
= VIy
= 0.56 ± 0.23. Thus we can state that even if the absolute values of y and v are uncertain, their ratio seems to be correct.
iQ3
io2 ~~
~
~
~~ ~
"r
~~, ~~
~
io~ lo' io~ io~ lo? io~
M~ / g mol~~
Fig. 7. Molar mass dependence of the hydrodynamic radius R~(6) and the radius of gyration R~(v).
Besides the mentioned exponents y, v and b two other exponents T and « are derived by percolation. These are the key exponents since they define the size distribution [3].
w(M) =AM~~~f(M/M*) (8)
with
M * (X~ X )~ ~'~ (9)
At the gel point the cut-off function f(M/M*) goes to unity, and a simple power law distribution should be observed. The two exponents T and « have in the FS and the lattice
percolation theories the values [3]
mean field percolation
T = 2.5 T
=
2,18
« =
0.5 «
= 0.45
The best method for measuring these two exponents would be a direct determination of the
distribution by the combined Size Exclusion Chromatography/Low Angle Laser Light
Scattering (SEC/LALLS) technique. Unfortunately, in our experiment the material remained adsorbed onto the gel, such that separation was not possible.
Another technique for determination of the size distribution results from the time correlation function (TCF), which is measured by dynamic light scattering [16, 17]. The TCF
gj(t) is a Laplace transform of the distribution of diffusion coefficients h(D ), defined as
ce
gj(t )
= h (D ) exp [- q~ Dt d In D (10)
o
where D is the translational diffusion coefficient. The inversion of this equation is a so-called
« ill posed problem », and its solution is involved with many difficulties. We applied the CONTIN program by Provencher [18, 19] to the correlation function at 20° of 3 different
samples with X near the critical region a) X
~ =
0.84 ; M~
= 2.6 x llJ~ g mol~
b) X~
=
0.87 M~ = 1.8 x 10? g mol~
c) X~
=
0. 89 ; M~
= 1.0 x 10~ g mol~ (sol-fraction)
where the last example (c) refers to a sample that was extracted from a gel. Power law behavior
h(D) ~D~ (11)
is observed in a short interval (Fig. 8). Within the limit of error the exponent was found to be
c = 0.35 ± 0.05.
~~o
iQ8
do
lo? q
°°°@g
q . o
-
lJ
C lO~ q ° ~°
~ °
O
i~~ o
lO~
io3
io~~~ io~~~ io~~~ io~~° io~ io~ io~? io~
o / ~~2~-1
Fig. 8. Distribution of the diffusion coefficient obtained by inverse Laplace transformation of
gj(t) at o
=
20° for X
= 0.84(o), 0.87(o) and 0.89(o).
It should be noted, that for a correct consideration the data have to be extrapolated to zero
angle, since at larger angle intemal modes of motion can contribute [16], and only at o =0 translational diffusion determines the relaxation spectrum and the TCF. Such
extrapolation is difficult, and a satisfactory procedure has not yet been found.
In order to convert the distribution of the diffusion coefficients into a molecular weight
distribution one can employ the D dependence on the molar mass for monodisperse particles
D
=
K'M ~° (12)
A molar mass dependence, however, could be established only with polydisperse samples
which is described by
D~ = KMj~ (13)
with K
=
4.35 x 10~~ mol cm~ s~ g~ and b
= 0.58 ± 0.03. This value for b agrees within the
experimental error with that from the molar mass dependence of R~.
The exponent b, however, differs from the required exponent a as a result of polydispersity
which increases with increasing M~. Assuming validity of the power law for the weight
fraction one finds for the relationship between a and b
a = b(3 T) (14)
since [20]
c = (2 T )la (15)
one obtains
T = (3 bc + 2)/(bc + Ii (16)
With the experimentally determined data for b and c we find T
=
2,17 ± 0.03 in agreement with three dimensional percolation and an exponent
u =
0.48 ± 0.04.
According to the Stokes Einstein relationship one has D
~ l/R~. The fractal dimension of
the individual clusters d~~ is defined by R~ -M~'~~ and with equation (12)
one find
lla
= (~
= 2,1±0.2. This dimension corresponds to the fractal dimension of a swollen, randomly branched cluster [3].
As mentioned before, G'(w) and G"(w) follow a power law at the gel point and
tan 3
=
G"(w)/G'(w) becomes independent of frequency. Percolation theory predicts a
power law exponent of n
=
0.67 in the limit of Rouse dynamics [2 Ii. Computer simulations, using an analogy of gelation to resistor-superconductor networks, gave n
= 0.73 [22, 23].
Mean-field theory predicts n = I.
Figure 4 shows master curves obtained by time-temperature superposition for a sample at X
=
0.87. Tan 3 still exhibits a frequency dependence and demonstrates, that the gel point
has not yet been reached, but the frequency dependence is much weaker than for the samples
at smaller X and at low frequency tan 3 is found to be tan 3
=
2.3 ± 0,I, corresponding to
n = 0.74±0.01. This exponent is in good agreement with the value obtained from the
extrapolation procedure shown in figure 5. Again the experimental exponent is closer to the value predicted by percolation than by mean-field theory.
6. Conclusion.
The main purpose of this paper was the determination of critical exponents in a gelling system
which occurred in a living crosslinking polymerisation. Evidence for percolation was found.
With regard of results obtained with other covalently [24-28, 6] and physically [29, 30]
crosslinked systems, we can state that in the critical region no influence of chemical
particularities or crosslinking mechanism can be observed.
A relevant result is the strong correlation between the critical exponents and the position of the gel point. We have chosen as critical value the point at which the corresponding exponent
shows a minimum error. In spite of large errors, the exponents found are those predicted by
three dimensional percolation theory.
Furthermore, the position of the gel point was crosschecked by oscillatory rheology, where power law behavior of G'(w and G"(w was taken as the criterion for the gel point.
Acknowledgements.
We thank H. H. Winter for helpful discussions and the opportunity to perform the rheological
measurements in Amherst. W. R, cordially thanks the Alexander von Humboldt-Stiftung for
a Feodor-Lynen Research fellowship. This work was done within the framework of the
Sonderforschungsbereich SFB60, supported by the Deutsche Forschungsgemeinschaft.
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