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Rheokinetics of curing of epoxy resins near the glass transition.


A study of rheokinetics of cure for polyfunctional oligomers is necessary for several important reasons. First, great changes in rheology and a fluid-to-solid state transition, at a certain degree of conversion, determine aspects of processing (choice of equipment, temperature and deformation regime of processing, and so on). Second, the performance characteristics (strength, deformability, residual stresses) of end products depend on the technology of the cure and are directly related to the kinetics of reactions and the corresponding evolution of rheology. Therefore, many important and interesting publications have been devoted to cure kinetics of epoxy resins (one of the most popular and typical polyfunctional oligomeric products) (1-4) and the transformation of the rheology (5-7).

There are two principal mechanical (other physical approaches are also used) methods to follow a process of cure. The initial stage of curing (gelation) as a rule is studied by a viscometric method. Indeed, the viscosity increase on cure is important both in theory and practice. The most convenient empirical equation for the time dependence of viscosity is an exponential-type (7, 8). However, this approach does not predict the existence of a threshold-gelation, where viscosity grows without limit. A theoretical-based equation is a relationship of a power type, which has been used in many publications (9-13). A theoretical value of the exponent is 0.7 (14), and this exact value was confirmed in some published results (15-17). However, in other papers it has been shown that the exponent value is not constant, but changes with temperature and composition (11-13). It was supposed that the deviation of the exponent from the universal value is a sign of the heterogeneity of a curing system (7, 11, 13).

The complete range of curing can be successfully investigated by various versions of a dynamic method when time dependences of elastic modulus and mechanical losses are measured (7). This approach is based on the concept of an ideal rubbery-like network model, and it is supposed that the elastic modulus is proportional to the network density. There are many publications in which this approach has been used (7, 18). However, the basic idea of this approach is valid when a curing system remains above [T.sub.g] during the whole process. Therefore the majority of rheokinetic works discuss the situation when an end product continues to be above [T.sub.g] (6), though engineering applications of cured resins are related to the glassy state. The most important exceptions are publications by Gillham et al. (19-23), where a so-called TTT-diagram is constructed and various possibilities connected with [T.sub.g] during (and as a result off curing are considered. One of these publications (23) demonstrated that the [T.sub.g] of the curing product (epoxy-amine system) can be used as an unambiguous measure of the degree of conversion (regardless the temperature of a reaction). In addition, it was shown that diffusion becomes rate-controlling for the curing process on approaching [T.sub.g]. This important result will be used in this discussion.

Of special interest is the interrelation between [T.sub.g] and the degree of conversion. The most fundamental result here is the following equation relating these parameters (24):

[T.sub.g] - [T.sub.g0]/[T.sub.g0] = [k.sub.1][Beta]/1 - [k.sub.2][Beta] (1)

where [T.sub.g] is an instantaneous [T.sub.g] depending on the degree of conversion, [Beta]; [T.sub.g0] is the [T.sub.g] of a linear polymer of the same chemical structure as after cure but without crosslinkings; and [k.sub.1] and [k.sub.2] are theoretical constants reflecting some physical properties of a polymer. As in Ref. 25, the [T.sub.g] of the initial reactive system will be assumed as [T.sub.g0].

Though the physics of the curing process near [T.sub.g] became clear after the works of the Gillham group, the influence of this relaxation transition on typical features of rheological transformation in oligomer curing and the possibility of its formal description remained open to discussion.

The aim of this paper is to provide experimental data on the rheokinetics of curing for four typical epoxy-based compounds and to propose a general explanation for these results.



The main object of investigation was epoxy-diane resin, ED-20. Its molecular mass is 390-420. Its viscosity is equal to 18.4 [Pa.sup.*]s (at 25 [degrees] C). The content of epoxy-groups is 19.9-22.0%. Curing agents used are listed in Table 1. Four different compositions were tested. They are listed in Table 2.
Table 1. Curing Agents and Their Main Characteristics.

 Viscosity Amine
 Chemical [Pa.sup.*]s at Number
Symbol Characterization 30 [degrees] C mg KOH/g

P-3 Reaction product between 2-3.5 205-225
 m-phenylene diamine
 and epoxy resin

T-4 Reaction product between 4-11 300-370
 hexamethylene diamine
 and epoxy resin

M-1 m-Phenylene diamine
 MM = 108
 [T.sub.m] = 62 [degrees] C
Table 2. Compositions and Their Destinations.

 Curing Agent
 ED-20, Mass Mass by
Destination by Parts Type Parts

I 100 T-4 80
II(*) 100 P-3 30
III 100 P-3 60
IV 100 M-1 14

* Stoichiometric ratio.


Rheological Methods

It is evident that there are two different stages of curing: In the beginning, we deal with a liquid, and after the gel point, the sample has lost its fluidity and becomes a solid. Thus it is convenient to use two different experimental methods: In the initial part, rheokinetics are well followed by a viscometric method, and the whole cure (but primarily its final part) can be monitored by measuring dynamic mechanical properties, storage modulus, G[prime], and loss tangent, tan [Delta].

Viscosity was measured in our experiments by a cone-and-plate rotational viscometer, Rheotest-2 (produced in Germany). The apparent viscosity of all samples did not depend on shear rate, and non-Newtonian flow was essentially absent. We suspect that non-Newtonian behavior may appear after micro-gelation, when local gel particles are formed and the system becomes heterogeneous (7, 26). In fact, we varied the shear rate during the process to reach the most reliable conditions for following cure.

Dynamic properties were measured by a free damping oscillation method using a standard frequency on the order of 1 Hz.

The gel point, [t.sup.*], was determined as the time when viscosity increased without limit. This point was found by an extrapolation of plots of dependences of reciprocal viscosity vs. time. Examples are shown in Fig. 1. One can see that the dependences drawn on this Figure are linear, and this confirms the method of extrapolation to [[Eta].sup.-1] [approaches] 0.


The curing process was also followed by a DSC method, by the following procedure. All components of a composition were mixed. Then, a 10-30 mg sample was placed inside an aluminum pan and capped. Isothermal curing was carried on at 25-80 [degrees] C. Then the pan was rapidly cooled and positioned into a calorimetric cell. After that, curing was continued by scanning, increasing temperature at the rate of 5 K/min from 10 to 180 [degrees] C. The heat of a reaction, AH, was measured as an area limited by a base line and a peak of the heat release curve. Thus we measured the residual enthalpy of reaction.

The total heat of a reaction of a sample, [Delta][H.sub.0], was also measured by the same procedure, but in this case an initial (mixed but uncured) sample, was placed into a calorimetric cell.

The "calorimetric degree of conversion," [[Beta].sub.c], was calculated as

[[Beta].sub.c] 1 - [Delta]H/[Delta][H.sub.0] (2)

The temperature dependences of all characteristic times and rate constants (introduced and discussed below) were treated in the coordinates of an Arrhenius equation. In all cases it appeared possible to find an apparent activation energy, E.


Viscosity Profiles

The initial parts of the isothermal time dependences of viscosity [ILLUSTRATION FOR FIGURE 2 OMITTED] for compositions I-III can be successfully described by an exponential equation:

[Eta] = [[Eta].sub.0] [(1 - t/[t.sup.*]).sup.-a] (3)

where [Eta] is current viscosity, [[Eta].sub.0] is initial viscosity, [k.sub.0] is viscosity-rate kinetic constant, and t is current time.

This equation is valid up to high viscosity levels, [approximately]500 [Pa.sup.*]s. This means that in many cases an exponential time dependence of viscosity can be used for practical calculations. An analogous equation was proposed and used for different curing systems (7, 8). However, this conclusion is not universal and the time dependence for our system IV cannot be described by the same kinetic equation ([ILLUSTRATION FOR FIGURE 2 OMITTED], curve 5).

Now let us analyze the final parts of the viscosity profiles. The best way to do this is to use so-called scaling theory (15, 16), which predicts the viscosity profile close to the gel point and is described by the following power-law equation

[Eta]/[[Eta].sub.0] = [(1 - t/[t.sup.*]).sup.-a] (4)

where a is the universal scaling factor theoretically equal to 0.7.

Several publications (14-17) demonstrate that this equation does work and the constant a is close to its theoretical value. Those works were conducted with specially selected specimens, which were characterized by a uniform (homogeneous) network. However, some contrary cases were also described (11-13). In these works the power a was arbitrary, though Eq 4 was valid. The reason for this deviation from the theoretical value was evident: heterogeneity of the cure, with microphase separation occurring before the gel point.

It is interesting to estimate the exponent values for our systems. They are presented in Fig. 3 as a function of temperature. One can see that any definite order in lay-out of experimental points is absent and any coincidence with the theoretical value can occur only by a happy chance. The reason for this is the same as in the former cases (7, 11, 13) - the heterogeneous nature of network formation in epoxy resins - and this is confirmed by different methods, including direct observations of structure (27).

However, it is reasonable to think that curing before micro-gelation proceeds in the same manner. Then several characteristic parameters, depending on temperature, must exist, and one can expect to construct universal curves reducing experimental data by these parameters. This idea was advanced for epoxy cure by Apicella et al. (6). They used the initial viscosity [[Eta].sub.0] of a reactive system and two "critical" values: time, t. (not the same as the gel time [t.sup.*]) and viscosity [[Eta].sub.*] as normalizing factors for constructing master curves of viscosity profiles for cure. This approach is used in treating our data in Fig. 4. One can see the possibility of constructing rheokinetic master curves. This demonstrates that for the initial stage of cure (at least), the same mechanism of a reaction takes place at different temperatures.

It is interesting to note that the initial rate of reaction and the gel-time are related by a simple relationship:

[k.sub.*][t.sup.*] = B

where B is a constant. Its values for the systems studied are as follows:
Composition B

I 5.06
II 4.68
III 1.95

An analogous relationship was also observed for other curing systems (26-28).

Dynamic (Viscoelastic) Properties

Use of a dynamic method for following a cure has certain advantages. Two are most important: first, it is a nondestructive method (small amplitudes of deformations); second, this method is applicable practically from the beginning to the end of cure.

Typical experimental data are presented in Fig. 5 showing two different sets. The systems studied pass through the [T.sub.g] if the curing process proceeds at 25 [degrees] C or 40 [degrees] C. It is proven by a two-peak character of tan [Delta]-vs-time dependence, as the first maximum reflects gelation and the second vitrification. At higher curing temperatures [T.sub.g] is not achieved.

There are some characteristic points in the dependence of dynamic properties on time. Loss maxima present both the gelation point and the [T.sub.g], and the final value of elastic modulus, [G.sub.[infinity]], at the end of cure. Various situations in the relative positions of transitions occurring during cure were discussed by Gillham et al. (19-21). The simplest case corresponds to the situation when the [T.sub.g] is not reached at the temperature of cure and the reaction proceeds to completion. Then the ratio

[[Beta].sub.r] = G[prime](t)/[G.sub.[infinity]] (5)

can be treated as a "rheological" degree of conversion (28) and [[Beta].sub.r] correlates with other measures of conversion, for example with the "calorimetric" degree of conversion (29).

The normalizing factor [G.sub.[infinity]] is evident if vitrification does not occur. But the situation becomes more complicated if vitrification takes place in the course of cure and the limiting degree of conversion is not reached. In the latter case, the limiting values of [G.sub.[infinity]] were obtained by extrapolation to lower temperatures via constants in kinetic equations, as described below.

The results of measuring G[prime](t) are the basis for a full rheokinetic description. It has been shown (7, 28) that in the case of complete cure, a rheokinetic equation of autocatalytic (self-acceleration) type is valid:

d[Beta]/dt = [k.sub.r](1 - [Beta])(1 + C[Beta]) (6)

where [[Beta].sub.r] is a rheological degree of conversion determined by Eq 5, [k.sub.r] is the initial rate of a reaction, and C is a characteristic constant not depending on temperature.

In some cases curing kinetics is better described by a second-order equation:

d[Beta]/dt = [k.sub.r][(1 - [Beta]).sup.2](1 + C[Beta]) (7)

In the case of incomplete cure (when and if a reaction is inhibited or restricted by diffusion limitations) a more complicated equation must be used (7):

d[Beta]/dt = [k.sub.r](1 - [Beta])(1 + C[Beta])(1 - b[Beta]) (8)

where b is a new constant representing the limiting degree of conversion, which equals [[Beta].sup.-1] (7).

Let us treat our experimental data in terms of Eq 6. The initial parts of the rheokinetic curves are presented in Fig. 6. Data are expected to be linearized in ln[[Beta]/(1 - [Beta]] vs. t coordinates. This appears true for I-III compositions (curing epoxy resins by amine adducts), as shown in Fig. 6.

Treating these linear graphs gives values of the kinetic parameters [k.sub.r] and C.

Data for the IV system are described by second-order kinetic, Eq 7, and again it allows us to find the value of the initial rate of reaction, [k.sub.r].

Equation 8 appears to be valid in all cases studied in this work, if cure leads to vitrification. Then it is reasonable to compare temperature dependences of a viscosity-rate constant [k.sub.0] and characteristic gel-times of the systems. This is done in Table 3, which also includes energy activations calculated from the results of dynamic measurements.

Extrapolation of dependences of [k.sub.r] to lower temperature ([T.sub.g] state) gives values of the constants that are used for calculating [Beta](t) dependence for temperatures when cure leads to vitrification.

It is also possible to compare results of kinetic studies made by dynamic and calorimetric methods. Calorimetric conversion for system I is also described by Eq 6 and the degrees of conversion found by the methods that coincide [ILLUSTRATION FOR FIGURE 7 OMITTED]. Then it is important that extrapolated (from dynamic measurements) values of [k.sub.r] appear to be quite the same as [k.sub.c] found by direct experiment.
Table 3. Activation Energies Calculated From Different
Experimental Data.

 Activation Energy, KJ/mol,
 Calculated According To:

Composition [t.sup.*] [k.sub.o] [k.sub.r]
I 44 44 -
II 36 36 29
III 43 43 41
IV 28 - 26

One experimental result can be discussed separately. As seen from Table 3, an increase in the content of the curing agent P-3 over the stoichiometric ratio results in marked difference in the rate constant and gel-time shortening. This can be explained by the different temperature dependence of the reaction rate of primary and secondary amino groups with epoxy groups (29, 30). This is the reason of increasing activation energy for a system with surplus curing agent.

Figure 7 gives gel-times found by a viscometric method, [Mathematical Expression Omitted] (as discussed above) and by a dynamic method, corresponding to the tan[Delta] maximum, [Mathematical Expression Omitted]. One can see that they are close to each other.

Now we shall try to understand the role of vitrification in the kinetics of cure. In the absence of vitrification, the process is described by the right-hand part of Eq 6, or by an analogous expression for the second-order kinetics. This value can be treated as a "chemical kinetic factor." In the initial part of the process, this factor coincides with the real reaction rate, i.e. the value d[Beta]/dt. But if a system approaches [T.sub.g], an observed reaction rate, d[Beta]/dt, appears less than the chemical kinetic factor. It has been determined by diffusion limitations at low temperatures and especially below [T.sub.g]. Then the effect of deceleration of a reaction can be discussed in terms of a "diffusion kinetic factor," [k.sub.d], which limits complete reaction. Indeed the overall reaction rate, k, can be expressed as:

[Mathematical Expression Omitted] (9)

Then if we know k and [k.sub.r] we can calculate [k.sub.d] by Eq 9. The procedure of calculation is as follows. We know [k.sub.r] from G[prime]([Beta]) measurements at the initial part of a reaction. It is assumed that this is the constant for the whole cure.

No one experimental method gives the possibility to find the degree of conversion close to the end of cure if a system is vitrifying. Then we used the Gillham method (22, 23) according to which a measure of cure is calculated from the [T.sub.g] by Eq 1.

The overall rate constant, k, can be calculated as the derivative d[Beta]/dt if we know the [Beta](t) dependence for the final part of reaction. The latter was found from [T.sub.g](t) data by Eq 1 and the values of [T.sub.g] were measured directly. To do this, we stopped the reaction at different moments, froze the system, and then, increasing temperature, found the glass transition point by the standard torsion pendulum method.

The result of [k.sub.d] calculations is shown in Fig. 8 as a function of temperature. Further, we try to describe temperature dependence of [k.sub.d] by the WLF-type equation as proposed by Gillham (21). It is reasonable, as the physical meaning of [k.sub.d] is close to physical values of parameters usually described by this equation. Then:

ln [k.sub.d] = ln [k.sub.d,0] + [Alpha](T - [T.sub.g])/[C.sub.1] - (T - [T.sub.g]) (10)

where [k.sub.d,0], [Alpha], and [C.sub.1] are empirical factors, though we can assume [C.sub.1] to be the universal constant [C.sub.1] = 51.6.

Figure 9 compares all the kinetic constants, k, [k.sub.r] and [k.sub.d], as a function of the degree of conversion, [Beta]. It is evident that the relative input of diffusion limitations becomes dominant on approach to [T.sub.g].


Processes of cure of four epoxy-amine systems were studied by methods of viscometry, dynamic mechanical analysis and scanning calorimetry. The investigation was carried on at temperatures below and above [T.sub.g] of completely cured products.

Viscosity growth in the initial stage of cure is described by an exponential-type equation, and on approach to the gel-point by a power law. However, the exponent in the power-law equation does not coincide with the universal theoretical scaling-law value.

Gel-times and activation energies of cure were found by different methods. It was established that the product of the initial reaction rate and the gel-time is constant at various conditions of cure. Time dependences of the degree of conversion calculated from calorimetric and dynamic mechanical data over the full range of conversions are the same and described by first- and second-order kinetic equations corrected by a factor representing the self-accelerating character of reaction.

If a reaction leads to a transition to the glass state in the course of cure, the self-acceleration equation is valid in the initial stage of the process only. The complete kinetic curve can be described by the DiBenedetto method relating the shift of [T.sub.g] with the degree of conversion. The rate constant for vitrificating systems is presented as a sum of reciprocal values of chemical and diffusion constants. Time and temperature dependences of kinetic and diffusion constants were calculated.


a = Scaling factor in Eq 4.

B = Empirical constant in Eq 9.

b = Constant in Eq 8 - a measure of incomplete curing.

C = Constant of self-acceleration in kinetic Eqs 6 and 7.

[C.sub.1] = Empirical constant in Eq 10.

E = Activation energy.

G[prime] = Elastic (storage) modulus.

[G.sub.[infinity]] = Normalizing value of storage modulus (modulus for a completely cured material in rubbery state).

k = Kinetic constant.

[k.sub.d] = Diffusion kinetic constant.

[k.sub.d,0] = Empirical factor in Eq 10.

[k.sub.r] = Chemical kinetic constant (initial rate of a reaction) in Eqs 6-9.

[k.sub.0] = Viscosity-rate kinetic constant in Eq 2.

[k.sub.1], [k.sub.2] = Constants in Eq 1.

[T.sub.g] = Glass transition temperature.

[T.sub.g0] = Glass transition temperature of an uncured sample.

t = Time.

[t.sup.*] = Gel-time.

[Mathematical Expression Omitted] = Gel-time found by a viscometric method.

[Mathematical Expression Omitted] = Gel-time found by a dynamic method.

[t.sub.*] = Normalizing time factor.

Greek Symbols

[Alpha] = Empirical constant in Eq 10.

[Beta] = Degree of conversion in curing.

[[Beta].sub.c] = Calorimetric degree of conversion according to Eq 2.

[[Beta].sub.r] = Rheological degree of conversion.

[Delta]H = Residual heat of reaction.

[Delta][H.sub.0] = Total heat of a reaction of uncured sample.

[Delta] = Angle of mechanical losses in dynamic measurements.

[Eta] = Viscosity.

k[[Eta].sub.0] = Initial value of viscosity before cure.

[[Eta].sup.*] = Normalizing viscosity factor.


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Author:Malkin, A. Ya.; Kulichikhin, S.G.
Publication:Polymer Engineering and Science
Date:Aug 1, 1997
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