Comparison of four methods for monitoring the kinetics of curing of a phenolic resin.
The kinetics of oligomer curing govern both the duration of the process and the properties of the final product. In fact, the meaning of the term "kinetics" is somewhat ambiguous because of uncertainty about which process is being described. If it is intended to mean the kinetics of network formation, there is still the question of whether the processes in question include chemical reactions of all kinds; the evolution of the average density of three-dimensional nodes, including or excluding closed loops; or the increase in the gel fraction. From an empirical point of view, "kinetics" may mean the rate of development of mechanical properties, but what kind of properties--viscosity, strength, rigidity, or something else? Thus, when discussing the kinetics of oligomer curing, it is necessary to first state the method used to monitor the process, and then it is of interest to know the relationship between the results obtained with different methods.
Differential scanning calorimetry (DSC) is the most popular method for monitoring oligomer curing [1-5], but it is well known that this method is not useful in the final stages of the reaction . It is sometimes supposed that the change of properties during the last stage of curing is related not to chemical reactions but to a relaxation that proceeds without thermal effects . However, an effect as large as an increase in glass temperature of 40K, as described by Ponomareva et al. , cannot be explained in terms of relaxation alone, without considering possible chemical transformations. It seems more likely that the chemical changes that occur during the final stage of the process are too small to be detected by standard DSC methods, but are still strong enough to affect properties that can easily be measured.
Rheological properties are useful for monitoring the curing of oligomers because of the important changes in these properties that occur during the process. In particular, the viscosity and modulus change by many orders of magnitude . However, chemical interpretation of the data is not straightforward. Dynamic mechanical analysis (DMA) became popular as a result of the work of Gillham  and Babayevsky and Gillham . This method is based on the theory of rubbery elasticity, which indicates that the rubbery modulus is proportional to the network density. However, this modulus describes behavior in the rubbery plateau, and it is difficult to ensure that measurements continue to be made in the rubbery plateau during the entire curing process, and especially near the end of the process. The DMA method then provides only a relative characterization, and the problem of quantitative interpretation remains. We conclude that the correlation of data from the various techniques is an open question.
Here we attempt to answer this question for the typical case of the curing of a phenolic resin by comparing the following methods: DSC, infrared spectroscopy (IR), viscometry, and DMA. In particular, we looked for correlations between the constants in the kinetic equations determined using these methods.
The reaction studied was the curing of phenol with a diisocyanate. The phenolic resin component consisted of phenol (70.5 g), paraform (30 g), zinc acetate (0.03 g), toluene (3.7 g), and dioctyl phthalate (22.5 g).
The phenol-formaldehyde oligomer reaction was carried at 130[degrees]C over a period of 4 hr, with water circulation through a Dean-Stark trap. The reactive mass was cooled to 30-35[degrees]C, and dioctyl phthalate and 0.05 ml of Pb[Cl.sub.3] were added. The reaction was stopped when the active phenol content fell to 1%. The product was a resol-type phenolic resin in the form of a light yellow viscous liquid.
The diisocyanate component was an 85% solution of 4,4'-di(phenyl methane diisocyanate) in toluene. The molecular weight was 250. Its other characteristics were as follows: melt temperature = 40[degrees]C; boiling temperature = 195[degrees]C (at 5 mmHg); freezing temperature = -38[degrees]C; and density ([d.sub.4.sup.50]) = 1.823g/[cm.sup.3]. The mass ratio of phenolic resin to diisocyanate was 55:45.
Rheological Properties. We measured the change in viscosity during oligomer curing using a Rheotest-2 rotational viscometer (Germany) with cone-and-plate fixtures. We measured the viscosity at several shear rates, and found that non-Newtonian effects were negligible up to the final stage of the reaction. Consequently, the shear rate is not reported below.
The storage modulus, G', and loss tangent, tan[delta], were measured using a torsion pendulum at a nominal frequency of 1Hz; a slight variation of frequency with temperature was neglected. The glass temperature, [T.sub.g], was taken to be the temperature at which tan[delta] had its maximum value.
Calorimetry. DSC measurements were carried out with a DuPont TA2000 thermograph. The time dependence of the heat output, Q(t), was measured at a temperature range of 25-60[degrees]C. When analyzing the DSC data it is important to establish the reference point, which is the total heat of the reaction, [Q.sub.max]. This was measured in the dynamic (scanning) mode at a temperature increase rate of 3K/min. The value [Q.sub.max] is the total heat of all reactions up to the completion of curing.
[FIGURE 1 OMITTED]
The degree of conversion in terms of the calorimetric degree of curing was calculated as follows:
[[beta].sub.cal](t) = Q(t)/[Q.sub.max]. (1)
Values of Q(t) were measured under isothermal conditions at several different temperatures.
IR Method. The IR method monitors the concentration of functional groups. The instrument used here was a Specord M-80 automatic spectrophotometer (Germany) with a chamber with NaCl windows. The change in optic density was measured at 2272 [cm.sup.-1], which is the band for the vibration of the C=N bonds in isocyanite groups. The density at 1604 [cm.sup.-1], which corresponds to the deformational vibration of the benzene ring, was selected as the inner standard . All measurements were made at 30[degrees]C.
RESULTS AND DISCUSSION
Figure 1 shows plots of viscosity versus time at three temperatures using both linear (Fig. 1a) and logarithmic (Fig. 1b) scales. Figure 1b shows that the viscosity evolution can be described by an exponential relationship:
[eta] = [[eta].sub.0][e.sup.[k.sub.[eta]]t], (2)
where [[eta].sub.0] is the initial viscosity, before the start of curing, which is weakly dependent on temperature, and k is a strongly temperature-dependent rheological kinetic constant. This type of relationship was reported previously [13-16] and has proven useful for practical applications.
This equation describes the data well, at least up to viscosities somewhat above [10.sup.3] Pa s. For practical purposes, one can define a "flow window" as the stage of reaction until the viscosity reaches some selected value (e.g., [10.sup.3] Pa s). However, this equation does not imply a gelation time t*, defined as the time at which the material no longer flows ([eta] = [infinity]).
A technique that has been used [17-19] to estimate a flow-based gel time is to plot the reciprocal of the viscosity versus time and extrapolate to find the value of t at which this quantity is equal to zero. When plotted in this manner, our data in the neighborhood of the gel point fell on a straight line, which implies that they obey Eq. 3:
[eta]/[[eta].sub.0] = (1 - t/t*)[.sup.-1]. (3)
We therefore estimated gelation times using this procedure.
Scaling theory suggests that a power-law expression should describe the dependence of viscosity on the time just before gelation [20, 21]:
[eta]/[[eta].sub.n] = (1 - t/t*)[.sup.-b], (4)
where b is a scaling factor equal to 0.7, and [[eta].sub.n] is a normalizing parameter. We found that our data could be approximated by this equation over a broad range of times, not just near the gelation point. However, the scaling factor b was not equal to the theoretical value, but was found to depend strongly on temperature, as shown below:
T, [degrees]C 25 30 35 45 60 b 1.07 1.29 2.08 2.82 4.35.
This type of behavior has been observed in several curing systems (e.g., Refs. 22 and 23), and it has been suggested that this departure from the scaling law arises from heterogeneity in the original oligomers .
Equations 2-4 describe three empirical dependencies of viscosity on time in a curing system, and it is of interest to look for relationships between the constants in these equations. First, Fig. 2 shows that there is a correlation between the rheological kinetic constant, [k.sub.[eta]] and the gel time, t*. This result can be described by the following relationship:
[k.sub.[eta]]t* = C. (5)
The line shown in Fig. 2 corresponds to C = 7.00 [+ or -] .05. This implies that for practical purposes, curing can be considered complete when [eta]/[[eta].sub.0] [congruent to] [10.sup.3].
[FIGURE 2 OMITTED]
The proportionality between the kinetic constant [k.sub.[infinity]] and the reciprocal of the gelation time 1/t* has been reported in the past for several different curing systems [16, 22-25]. It thus appears that Eq. 5 has general applicability to curing processes. However, Eq. 5 can only be valid if the apparent energies of activation for [k.sub.[infinity]] and t* are the same. The values of these activation energies, calculated from our data using the Arrhenius equation, are as follows:
from the [k.sub.[eta]](T) dependence: 53 KJ/mol
from the t*(T) dependence: 44 KJ/mol.
The two values are indeed reasonably close to each other.
The calorimetric degree of conversion [[beta].sub.cal], defined in Eq. 1, is shown as a function of time in Fig. 3. We note that the limiting degree of conversion [[beta]*.sub.cal] increases with temperature. This type of behavior was observed in previous studies [26-28]. It has been proposed that the time dependency of [[beta].sub.cal] at any temperature can be described by the following expression [27, 28]:
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
[d[[beta].sub.cal]]/dt = [k.sub.cal](1 + C[[beta].sub.cal])(1 - [[beta].sub.cal])(1 - [zeta][[beta].sub.cal]) (6)
where [k.sub.cal] is a strongly temperature-dependent calorimetric kinetic constant that is equal to the initial rate of a reaction. The factor involving C describes self-acceleration, and the one involving [zeta] accounts for self-inhibition and the existence of a limiting degree of curing, since at [[beta]*.sub.cal] = [[zeta].sup.-1] the rate of the reaction becomes zero. The applicability of this equation to curing processes with incomplete conversion has been demonstrated for several systems [24, 28, 29], and it also describes our data quite well.
The apparent activation energy determined from the dependence of [k.sub.cal] on temperature was about 58 kJ/mol. This value is close to the above-cited values of the activation energies inferred from the [k.sub.[eta]](T) and t*(T) data. Also, we found that C was about 27 and did not depend on temperature.
Figure 3 shows that complete calorimetric conversion, [[beta]*.sub.cal] = 1, was not achieved at any temperature. Values of [[beta]*.sub.cal] were found to be 0.5 in the range of T between 35[degrees]C and 45[degrees]C, 0.58 at 55[degrees]C, and 0.8 at 60[degrees]C. The most likely reason for this is the vitrification of the system at a certain degree of curing as can be clearly seen in the well-known TTT (Time-Temperature-Transformation) diagram [10, 11]. The cessation of the curing process occurs above the glass transition temperature due to the formation of a dense network that inhibits mobility and contacts between reactive groups . In the present case, [[beta]*.sub.cal] is reached below the glass transition temperature, as demonstrated by the low value of the modulus at the time corresponding to [[beta]*.sub.cal]. In addition, there is no maximum in the loss tangent at this time (see the discussion about DMA data below).
Chemical Group Transformation
Figure 4 shows our IR data, which are well described by a second-order kinetic model:
[dot.[beta].sub.IR] = [k.sub.0,IR](1 - [[beta].sub.IR])[.sup.2] (7)
where [[beta].sub.IR] is the relative concentration of reactive C=N bonds, which decreases during curing, and [k.sub.0,IR] is the initial rate of a reaction, as determined from spectroscopic data. At 200[degrees]C, [k.sub.0,IR] is 0.065 [min.sup.-1]. It is obvious that Eqs. 6 and 7 and not equivalent, so it is unlikely that the constants [k.sub.cal] and [k.sub.0,IR] would be related. This means that for the system studied, calorimetric and spectroscopic data are not equivalent.
Correlation of Viscometric and Calorimetric Data
Now we consider the relationship between viscosity and the calorimetric degree of conversion during the curing reaction. According to percolation theory [20, 21], near the gel point, this relationship is given by
[eta]/[[eta].sub.m] = ([beta]*/[[beta]* - [beta]])[.sup.-s] (8)
where [[eta].sub.m] is an empirical parameter, and s is a scaling factor. A somewhat more general expression has also been proposed [17, 31].
[eta]/[[eta].sub.m] = ([beta]*/[[beta]* - [beta]])[.sup.F([beta])] (9)
Kim and Kim  assumed that F([beta]) is a linear function:
F([beta]) = a + b[beta]. (10)
Where a and b are empirical constants.
Khusid and Vlasenko  proposed the following relationship between [eta] and [beta]*:
[eta]/[[eta].sub.0] = exp[[k.sub.c]([beta]*/[[beta]* - [beta]])] (11)
where [k.sub.c] is an empirical coefficient. It is interesting to note that the three empirical equations (Eqs. 8, 9, and 11) all involve the group [([beta]*/([beta]* - [beta])], where [beta] is understood as [[beta].sub.cal]. More complex equations for [eta]([beta]) have been proposed, but they are not useful for practical applications because they include too many empirical constants or physical constants that cannot be reliably evaluated.
Our data did not follow any of the above equations; we observed that the viscosity increased very sharply near the gel point, and that for practical purposes the viscosity development could be modeled as follows:
[eta](t) = [[eta].sub.0] for t < t*
[eta](t) = [infinity] for t [greater than or equal to] t*.
This simple approximation is useful in modeling the flow of curing liquids [34, 35], as it reflects the principal features of the variation of viscosity during curing.
[FIGURE 5 OMITTED]
Equations 8, 9, and 11 imply that [eta] [right arrow] [infinity] as [beta] [right arrow] [beta]*. However, the viscometric gel time t* is not the same as the calorimetric limiting time [t*.sub.cal]. Figure 5a demonstrates this by comparing these two times, and we see that t* < [t*.sub.cal]. Figure 5b compares the limiting degree of cure determined directly from calorimetric data with that corresponding to t*, the viscometric gel point time. This means that this last quantity, [beta](t*) [equivalent to] [[beta]*.sub.visco], and not [[beta]*.sub.cal], should be the limiting value in an equation for [eta]([beta]). We also note that the viscometric gel point can depend on temperature, as shown by Han and Lem .
The use of the storage modulus G'(t) at a constant frequency to monitor curing reactions is based on rubber elasticity theory, in which the elastic modulus is proportional to the molecular network density. However, the assumptions of rubber elasticity theory are not valid for the majority of DMA experiments. Therefore, DMA data provide only a relative measure of curing that may not be quantitatively related to chemical or structural transformations.
The tracking of curing kinetics at high degrees of conversion can only be carried out using DMA, because calorimetric and chemical methods lose their sensitivity at this stage of the process. This can be illustrated by the use of a simple model. Let a reactive mass contain N active particles reacting at their ends. Then, at the z-th step of the reaction, the thermal effect is Q = (N/2z), i.e., Q [proportional] [z.sup.-1], and at high z values Q becomes very small. Meanwhile, the same reaction leads to an increase in chain length and, consequently, to a linear increase in the modulus, which is thus roughly proportional to z.
It has sometimes been reported that there is a direct correlation between calorimetric and DMA data [16, 36, 37], but our findings do not support this idea. A comparison of our calorimetric and DMA data shows that important changes in the modulus and loss tangent occur long after thermal effects have ceased. It is thus obvious that these two methods respond to different stages of the process. Moreover, if we assume that the maximum in tan[delta] corresponds to vitrification, it is evident that curing, as indicated by the increase of the modulus, continues after this transition takes place. Similar results were obtained at other temperatures.
The time dependency of the storage modulus indicates incomplete curing, and the final values of G' (as t [right arrow] [infinity]) appear to be temperature-dependent. This is typical behavior for curing systems [24, 27-29].
Our next step was to characterize the degree of curing throughout the entire process. One possibility is to express [[beta].sub.DMA], the degree of conversion as measured by DMA, as follows:
[[beta].sub.DMA] = [G(t) - [G.sub.0]]/[[G.sub.[infinity]] - [G.sub.0]] (12)
where G(t) is the current value of the storage modulus at a given frequency, [G.sub.0] is its initial value, and [G.sub.[infinity]] is its final value. As a general rule, [G.sub.0] [much less than] [G.sub.[infinity]], and [[beta].sub.DMA] can be expressed as follows:
[[beta].sub.DMA] = [G(t)]/[G.sub.[infinity]]. (13)
The time dependence of [[beta].sub.DMA] can be described by a standard kinetic expression. For the system of interest here, there is no induction period, and it is possible to fit the data with a second-order kinetic equation:
[d[[beta].sub.DMA]]/dt = [k.sub.DMA](1 - [[beta].sub.DMA])[.sup.2]. (14)
Although the forms of the kinetic expressions for calorimetric data (Eq. 6) and DMA data (Eq. 14) are different, it is of interest to compare the initial rates of the reaction (i.e., the values of [k.sub.cal] and [k.sub.DMA]). This is shown in Fig. 6, and as can easily be predicted, there is no agreement between the two rates.
Thermomechanical Analysis--High Degrees of Conversion
The use of the DMA method (e.g., Eq. 14) for treating the experimental data would be acceptable if the whole process of curing proceeded in the rubbery state, and [G.sub.[infinity]] had the meaning of the equilibrium rubbery modulus, which does not depend on curing temperature. However, as noted above, this is not true for the system studied here, or for many other systems for which data have been reported.
[FIGURE 6 OMITTED]
Data that have a sounder physical basis are to be expected if one characterizes the degree of curing by tracking the glass temperature, [T.sub.g], when the material has a relaxation transition during the curing process. The equation proposed by DiBenedetto  relates the glass temperature to the degree of curing as follows:
[[T.sub.g] - [T.sub.go]]/[T.sub.go] = [[C.sub.1]([T.sub.g] - [T.sub.go])]/[[C.sub.2] + ([T.sub.g] - [T.sub.go])] (15)
where [C.sub.1] and [C.sub.2] are constants, [T.sub.g0] is the initial glass temperature, and [T.sub.g]([beta]) is the current glass temperature, which depends on the degree of curing. This equation can be rearranged in a linear form to facilitate the reporting of experimental data:
[T.sub.g0]/[[DELTA][T.sub.g]] = [a/[beta]] - b (16)
where a and b are constants, and [DELTA][T.sub.g] = [T.sub.g] - [T.sub.g0].
Experimental data presented in terms of the quantity on the left in Eq. 16 are shown in Fig. 7, where [T.sub.g] was determined from the maximum in tan[delta], and [beta] is the degree of curing as found by the calorimetric method, [[beta].sub.cal]. We see that Eq. 16 does describe the data and can be used to determine [beta] by measuring the development of the glass temperature at high degrees of curing.
Finally, we note that our observations agree with those of previous studies [22, 24, 39, 40] regarding several key points:
1. The exponent in the scaling equation (Eq. 4) has a variable value.
2. The values of [beta]* are temperature-dependent.
[FIGURE 7 OMITTED]
3. Gel particles and fractions appear before the entire sample gels (i.e., separation of micro- and macrogelation).
4. The kinetics of curing are independent of concentration of the reactive mass when curing takes place in a solution.
5. The rate of curing is dependent on the shear rate after the point of microgelation.
These experimental observations cannot be explained by the geometrical model of statistical curing, and provide evidence of the heterogeneous nature of the process. Therefore, the kinetic equations discussed above are of interest primarily for fitting experimental data obtained by various experimental methods.
Viscosity growth in phenolic resin curing can be described by an exponential equation, and the rate of viscosity increase can be described in terms of a "viscometric" kinetic constant, [k.sub.[eta]]. A gel time, t*, was found by extrapolating the time dependence of the reciprocal of the viscosity. The product of t* and the viscometric kinetic constant [k.sub.[eta]] is constant for all temperatures, and this result is valid for phenolic resin compositions as well as for many other curing systems.
The kinetics of curing as monitored by different methods are described by very different fitting equations. "Calorimetric" kinetic data are described by a kinetic equation that includes the effects of self-acceleration and self-inhibition (incomplete curing).
The gel time found by the viscometric method is different from that corresponding to completion of curing as found by the calorimetric method.
Data from the IR method are well approximated by a second-order kinetic equation.
The degree of conversion at the gel point depends on temperature. It is supposed that this phenomenon (as opposed to the model based on homogeneous curing) reflects the heterogeneity of the curing process.
The kinetics of curing at high degrees of conversion can be followed by the DMA method. The change in the elastic modulus can be described by a second-order kinetic equation, although its kinetic constant is quite different from the kinetic constant based on thermal effects.
The glass temperature (determined by the maximum in the loss tangent) is related to the degree of curing by the DiBenedetto equation.
The experimental data demonstrate that the several methods commonly used to monitor curing processes reflect quite different processes and give noncomparable results.
a, b: empirical constants in several equations b: scaling factor C: constant describing the effect of self-acceleration [C.sub.1], [C.sub.2] constants in DiBenedetto equation G: storage modulus at 1 Hz G(t): current value of modulus [G.sub.0]: initial value of modulus G[infinity]: final value of the modulus or the equilibrium rubbery modulus [k.sub.c]: empirical coefficient [k.sub.cal]: "calorimetric" kinetic constant [k.sub.[eta]]: "rheological" kinetic constant [k.sub.0.IR]: initial rate of reaction as determined from spectroscopic data [t.sub.cal]: gel time corresponding to [[beta]*.sub.cal] t*: gel time as found by the viscometric method tan[delta]: loss tangent T: temperature [T.sub.g0]: initial glass temperature [T.sub.g]([beta]): current glass temperature that varies with degree of curing [beta]: degree of conversion (determined by any method) [[beta].sub.cal]: "calorimetric" degree of conversion [[beta]*.sub.cal]: limiting "calorimetric" degree of conversion [[beta].sub.DMA]: degree of conversion as measured by the DMA method [[beta].sub.IR]: relative concentration of reactive C=N bonds [[beta]*.sub.visco]: degree of conversion corresponding to t* [UPSILON]: constant describing the effect of self-inhibition [[eta].sub.n]: normalizing parameter [eta]: viscosity [[eta].sub.0]: initial viscosity (before the start of curing)
The authors are grateful to Mr. I.N. Balashov and Mr. S.I. Kazakov for their assistance in the experimental part of this work. We also thank Professor J. Dealy, (McGill, Montreal), who read the text, made valuable comments, and helped us improve the English.
Presented in part at the 6th European Conference on Rheology, Erlangen, Germany, 2002.
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Institute of Petrochemical Synthesis, Russian Academy of Sciences, Moscow, Russia
I.Yu. Gorbunova, M.L. Kerber
Mendeleev Institute of Chemical Technology, Moscow, Russia
Correspondence to: A.Y. Malkin; e-mail: email@example.com
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|Author:||Malkin, A.Ya.; Gorbunova, I.Yu.; Kerber, M.L.|
|Publication:||Polymer Engineering and Science|
|Date:||Jan 1, 2005|
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