Monitoring the reaction progress of a high-performance phenylethynyl-terminated poly(etherimide). Part I: cure kinetics modeling.
Application of fiber-reinforced polymeric composites (FRPCs) are limited by their relatively poor performance in high-temperature conditions. To take full advantage of these high specific strength materials, processable polymers with high temperature stability have been in development (1). Synthesis of a material consisting of an Ulterm[TM]-type poly(etherimide) with a 4-phenylethynylphthalic anhydride endcapping agent was successful In producing a melt-processable, high-glass transition temperature (250[degrees]C-260[degrees]C), sufficiently ductile network material with good thermal-oxidative stability, and solvent resistance (2). This phenylethynyl-terminated Ultem[TM] (PETU) was developed with high-performance adhesive and composite applications in mind; the structure of PETU is shown in Fig. 1. Promising results were obtained from previous investigations of the properties of carbon fiber-reinforced PETU composites (3-7). The intended applications of this material dictate that it must survive constant m oisture sorption due to fluctuations in service temperature and relative humidity, long-term exposure to high temperature (~177[degrees]C), and attack by solvents.
Thermal, oxidative, hydrolytic, and solvent-induced degradation of thermosetting polymers often are accelerated by lack of complete cure; unreacted monomers may react with penetrants to weaken the matrix. The presence of unreacted monomers also may reduce the density of the matrix and thus accelerate diffusion of penetrants into the matrix. A consolidation and cure schedule that minimizes remaining unreacted monomer in a composite part is desired. Therefore, a model must be developed that accurately predicts degree of cure as a function of time and temperature.
There is a consensus among researchers that the cure kinetics, mechanisms, and resultant cured structure of phenylethynyl-terminated (PET) monomers/oligomers are not well understood (8-17). An understanding of the cure mechanism and cured structure is necessary for theoretical predictions regarding the time and temperature effects on cure, long-term properties, and durability. However, the scope of this research is limited to developing an appropriate cure schedule for 2500 g/mol (2.5k) PETU, which may be satisfied with empirical models of cure kinetics. Therefore, this review is intended to gather available information concerning the reaction of PET mers to provide a background for critiquing results of thermal analysis of 2.5kPETU. The information summarized here is from results of differential scanning calorimetry (DSC) and Fourier transform infrared spectroscopy (FTIR). Both techniques are able to provide insight into the reaction kinetics of PET cure. In addition, the differences in the mechanisms by whi ch each technique monitors reaction progress also provides information about reaction kinetics.
All possible reaction paths for PET mers consume the acetylene groups (8). Therefore reaction progress can be observed via FTIR by monitoring the intensity of the infrared absorption at the acetylene wave number, approximately 2212 [cm.sup.-1] (8). Several other studies (9-13) confirm that the carbon-carbon triple bonds are consumed with reaction progress and have monitored extent of reaction via FTIR measurements.
Tan (9) investigated the cure of 2kPETU and 3kPETU at temperatures ranging from 300[degrees]C to 345[degrees]C and reports that the disappearance of the triple bond appeared to follow first-order reaction kinetics for the initial 80 minutes of cure. After 80 minutes, the triple bonds reportedly were consumed faster than dictated by first-order reaction kinetics. From these FTIR results, activation energies ([E.sub.a]) of 114 [+ or -] 10 kJ/mol and 75 [+ or -] 10 kJ/mol were calculated for 3kPETU and 2kPETU, respectively.
The FTIR results were supplemented by dynamic DSC measurements from which Tan (9) calculated [E.sub.a]s of 139 kJ/mol and 124 kJ/mol for 3kPETU and 2kPETU, respectively, via the Ozawa (18) method. The [E.sub.a]s determined for several other similar PET oligomers demonstrated that [E.sub.a] increased with increasing molar mass, suggesting diffusion plays an important role in the curing process.
Wood, et al. (11) studied a model PET compound containing a single ethynyl bond: 4-phenoxy-4'phenylethynylbenzophenone having a molecular weight of 374 amu. The carbon-carbon triple bond FTIR band was greatly diminished following a 15 minute hold at 37500, indicating reaction was near completion and/or the remaining concentration of triple bonds was below the instrument's detection limit. Takekoshi and Terry (15) synthesized two model PET compounds: 1) N-(3-phenylethynyl) phthalimide (N3) and 2) N-phenyl[4-(phenylethynyl)phthalimide] (N4); each containing a single acetylenic group. The N3 compound displayed second-order reaction kinetics with an [E.sub.a] of 132 kJ/mol. The N4 compound displayed first-order reaction kinetics with an [E.sub.a] of 153 kJ/mol.
A promising PET imide with a molecular weight of 5000 g/mol has been developed at the NASA Langley Research Center and is appropriately named LaRC PETI-5 (14). A DSC study utilizing the isoconversional plot technique of Flynn, Wall, and Ozawa reports that all stages of the cure reaction of PETI-5 can apparently be described with an average [E.sub.a] of 139.0 [+ or -] 3.3 kJ/mol (14). In contrast to some of the other studies mentioned here, which report first-order reaction kinetics for at least part of the cure of PET oligomers, Hinkley (14) reports that the reaction cannot be described by a simple order even though the temperature dependence of the reaction can reportedly be described with a single activation energy.
Sastri et al. (8) studied the cure of a PET monomer by synthesizing 1, 2, 4-tris(phenylethynyl-benzene) with a molar mass of 384 g/mol. These FTIR results indicate first-order reaction kinetics with an [E.sub.a] of 135 kJ/mol. Isothermal and dynamic DSC were also utilized to monitor reaction advancement. In general, the degree of cure determined from residual exotherms measured following isothermal holds did not correlate well with the degree of cure calculated from intensities of the acetylenic FTIR bands following identical isothermal holds. In contrast, the [E.sub.a] (137 kJ/mol) determined via the Ozawa method did agree well with the FTIR determined [E.sub.a]. However, the Ozawa method is not adversely affected by the diffusion control that apparently takes over the reaction at later stages. Evidence of diffusion control is apparent in the FTIR results, which display an initial steep rise in conversion versus time followed by a transition to a gradual increase in conversion. The lack of correlation betwee n DSC residual exotherms and remaining FTIR band intensity is further evidence of diffusion control taking over the reaction. When the reaction becomes diffusion-controlled, FTIR is much better suited than DSC to measure the fraction of unreacted material. For DSC to measure a residual exotherm, the remaining unreacted material must react. If the reaction has progressed to the point that the diffusion necessary for further reaction is severely restricted, a residual reaction cannot occur in the time span of a DSC scan and thus a residual exotherm is not detected. However, FTIR does not require reaction, and consequently diffusion, to measure the concentration of unreacted material. Therefore, FTIR is able to detect unreacted material that is undetectable via DSC.
CURE KINETICS MODELING BACKGROUND
The cure of 2.5kPETU was analyzed via DSC in order to develop an empirical kinetics model to assist in the selection of a cure schedule for 2.5kPETU. In addition, there was a desire to compare the kinetics of 2.5kPETU with those of other PET imides. Reaction kinetics studies begin with a basic rate equation that correlates the reaction rate, d[alpha]/dt, to the concentration of remaining reactants, (1 - [alpha]), through a rate constant, k (Eq 1) (19). In the case of PETU, the remaining reactants are the phenylethynyl groups with one or two available reactive sites, and the concentration of the remaining reactants may be determined from the degree of cure, [alpha].
d[alpha]]/dt = k(T)f([alpha]) (1)
Two general categories describe most thermoset cures: nth-order and autocatalytic. Kinetics of nth-order assume that once a reactant has reacted, it will not contribute to further reaction. As a result, the reaction rate is proportional to the concentration of unreacted material (19). Therefore, only the unreacted material has to be considered in determining the reaction rate, as shown in Eq 2, where n is the reaction order.
f([alpha]) = [(1 - [alpha]).sup.n] (2)
During isothermal cure of an autocatalytic material, the reaction rate accelerates until a maximum is reached at 20% to 40% of full cure. The acceleration is a result of reaction products aiding further reaction. The reaction scheme for PET mers is unknown and there are several proposed schemes (11), but in short, the breaking of an ethynyl bond and creation of a new bond may aid additional reaction at that site or a nearby site. The dependence of reaction rate on degree of cure for an autocatalytic reaction is shown in Eq 3, where the superscripts m and n are reaction orders.
f([alpha]) = [[alpha].sup.m][(1 - [alpha]).sup.n] (3)
In both autocatalytic and nth-order reactions, the rate constant, k, carries the temperature dependence of the reaction through an Arrhenius relationship given in Eq 4 (19).
k(T) = Z exp[-[E.sub.a]/RT] (4)
Most thermoset reactions can be sufficiently represented by one of these chemically based models, which implies chemical reactions control the cure rate. Two notable exceptions exist. One exception is the case of diffusion control, which often replaces chemical reaction control as a material vitrifies and becomes glassy. Considerations of diffusion control will be addressed later. The other exception occurs when more than one type of chemical reaction is occurring. In this situation, one reaction may be nth-order and the other autocatalytic. This combination of reactions may be represented by Eq 5, or by a simpler form--Eq 6. This combination model has been used in many cases where the reaction rate peaks after the beginning of the reaction, but the initial reaction rate ([k.sub.1]) is nonzero (20 and references within). The combination reaction kinetics model given in Eq 5 and 6 worked well in the modeling of isothermal cure of 2.5kPETU, as will be described later.
d[alpha]/dt = [Z.sub.1] exp[-[E.sub.a1]/RT [(1 - [alpha]).sup.n] + [Z.sub.2] exp [-[E.sub.a2]/RT [[alpha].sup.m][(1 - [alpha]).sup.n] (5)
d[alpha]/dt = ([k.sub.1] + [k.sub.2][[alpha].sup.m][(1 - [alpha]).sup.n] (6)
A DSC technique based mostly on isothermal measurements was used to determine the kinetic parameters to describe the cure of 2.5kPETU. Prior to beginning thermal analysis, 2.5kPETU powder was degassed under vacuum at 165[degrees]C for 4.5 days to remove residual o-dichlorobenzene remaining from synthesis. At ambient pressure, the boiling point of o-dichlorobenzene is 179[degrees]C. However, heating the powder above 170[degrees]C results in partial fusion of the powder. Under vacuum, 165[degrees]C is adequate to evaporate the o-dichlorobenzene. DSC specimens were prepared by sealing 5 to 6 mg of degassed powder in hermetic aluminum pans. DSC experiments were done in a TA Instruments DSC 2920 with a 35 ml/min nitrogen purge. The DSC experiments commenced with three 5[degrees]C/min ramps from 50[degrees]C to 425[degrees]C; degradation appeared to occur if ramps exceeded 425[degrees]C. These ramps displayed a consistent exotherm onset temperature of 325[degrees]C, peak temperatures ranging from 368[degrees]C to 3 72[degrees]C, and a total heat of reaction, [DELTA][H.sub.T], of 112.5 J/g (the area of the peak was calculated using a sigmoidal baseline). The glass transition temperature, measured at the inflection point, of the uncured 2.5kPETU was approximately 190[degrees]C. One of these ramps is displayed in Fig. 2.
Isothermal measurements were run at 5[degrees]C increments ranging from 315[degrees]C to 390[degrees]C. The shapes of d[alpha]/dt versus [alpha] curves displayed three distinct temperature regions in which the cure kinetics appeared to differ: T < 325[degrees]C, 325[degrees]C [less than or equal to] T [less than or equal to] 360[degrees]C, and T > 360[degrees]C. Because previously recommended cure schedules for PETU materials utilized holds at 350[degrees]C the isothermal measurements from 325[degrees]C to 360[degrees]C were selected for the production of the isothermal data necessary for the development of a cure kinetics model. In addition, this temperature range covers the central portion of the frontside of the exotherm peak, where the reaction rate is increasing the most rapidly with temperature increase.
Isothermal measurements were performed by: preheating the closed DSC cell to the desired temperature, opening the cell and quickly placing the specimen in the nitrogen purged cell, allowing the cell temperature to recover to within 4[degrees]C of the set temperature, and then beginning data collection (21).
After each specimen went through an isothermal measurement, it was removed from the DSC cell and allowed to cool. The specimen was then returned to the DSC cell for a 5[degrees]C/min ramps from 150[degrees]C to 425[degrees]C to measure residual cure and the [T.sub.g] resulting from each isothermal measurement.
RESULTS AND DISCUSSION
As with the temperature ramp shown in Fig. 2. the isothermal holds produce a measurement of heat flow that monitors reaction progress. However, the isothermal measurements monitor heat flow versus time rather than temperature as shown in Fig. 3a. The area under the heat flow curve represents the total heat of reaction for the isothermal hold, [DELTA][H.sub.I]. Thus, the selection of boundaries for this area is important. A time must be selected as the start of cure, [t.sub.s], and a time must be selected as the finish of detectable cure, [t.sub.f]. The traditional method for selecting [t.sub.s] and [t.sub.f] begins by placing [t.sub.f] at the first local minimum along the heat flow plateau (Fig. 3a). A baseline is then constructed with the baseline heat flow equal to the reaction heat flow at [t.sub.f] (Fig. 3a). The location of [t.sub.s] is then moved until the sum of [DELTA][H.sub.I] and [DELTA][H.sub.R], the residual heat of reaction measured via a temperature ramp after the isothermal measurement, is equi valent to [DELTA][H.sub.T], the total heat of reaction, from the temperature ramp on virgin material. However, the majority of the residual ramps in this study displayed negligible or no [DELTA][H.sub.I] despite different values of [DELTA][H.sub.I]. Sastri et al. (8) saw the same lack of residual exotherms. The lack of a residual exotherm suggests the reaction has progressed into the diffusion control region as discussed previously. Thus a different method for selecting [t.sub.s] was necessary.
The majority of the isothermal measurements demonstrated a small, local minimum within the overall heat flow maximum at the beginning of the hold as seen in Fig. 3b. This local minimum was believed to be near the transition from DSC cell equilibration to energy output from the actual exothermic reaction. Thus, this point was selected as [t.sub.s] for all isothermal measurements. Most likely, the reaction begins prior to [t.sub.s], but the local minimum was the most consistent landmark for choosing [t.sub.s] from measurement to measurement. Some error was likely incurred from inconsistent amounts of reaction occurring prior to [t.sub.s] and being omitted from the data collection.
Following the selection of [t.sub.s], [DELTA][H.sub.I] was calculated for each isothermal measurement (Table 1). As would be expected for isothermal holds, all eight temperatures have [DELTA][H.sub.I] values less than [DELTA][H.sub.T]. There appeared to be no general trend between hold temperature and [DELTA][H.sub.I], except at the highest temperatures where [DELTA][H.sub.I] decreased with increasing temperature due to greater portions of the initial reaction being missed during cell equilibration. Because of the lack of a trend between hold temperature and [DELTA][H.sub.I] it was difficult to estimate the amount of reaction enthalpy that was being missed during cell equilibration and the amount that was not detectable at the end of cure. Therefore, using [DELTA][H.sub.T] calculate the isothermal degree of cure, [[alpha].sub.I], for each temperature would have resulted in an inaccurate prediction of [[alpha].sub.I] at the completion of detectable cure. Furthermore, since the ultimate objective of this study was to develop the ability to determine the necessary processing times for any specified isothermal cure temperature within the experimental temperature range, using the [DELTA][H.sub.I] values for each individual isothermal cure temperature provided for better prediction of the time required to reach the completion of detectable cure for a specified temperature. Knowing [DELTA][H.sub.I], [[alpha].sub.I] could be calculated at any intermediate time, [t.sub.i] between [t.sub.s] and [t.sub.f]. Calculation of [[alpha].sub.I] required knowing [DELTA][H.sub.I](t), the heat evolved between [t.sub.s] and [t.sub.i] as shown in Fig. 3b. The value of [[alpha].sub.I] was then simply [DELTA][H.sub.I](t)/[DELTA][H.sub.I]. Thus at any time, the experimental value of [[alpha].sub.I] could be calculated.
Several sources (8, 9, 15) reported first-order reaction kinetics for PET monomers/oligomers. Having the ability to determine the experimental value of [[alpha].sub.I] at any time allowed for the investigation of whether first-order reaction kinetics may be applicable to 2.5kPETU. To make this determination, a relationship between [[alpha].sub.I] and time was derived to determine the first-order reaction kinetic parameters [E.sub.a] and Z. Equation 7 is generated by combining Eqs 1, 2, and 4 and then integrating.
ln(1 - [alpha])= - k(T)t (7)
Subsequently. ln(l - [alpha]) may be plotted versus time for each isothermal cure temperature with the slope of a linear fit providing the negative of the rate constant for each temperature (Fig. 4). The plot of each temperature was fit linearly up to the point that the slope transitioned to an almost vertical relationship between ln(1 - [alpha]) and time, or in the case of 355[degrees]C, the slope leveled (fits are not shown). Details regarding the fits are given in Table 2. All fits except the one for 35500 were beyond [[alpha].sub.I] = 0.99.
The high correlation coefficients indicate first-order kinetics may apply to the reaction of 2.5kPETU oligomers. Recalling from Eq 4 that the rate constant, k, is a function of temperature and depends on the activation energy, [E.sub.a], the natural logarithm of Eq 4 may be utilized to determine a single activation energy for 2.5kPETU (Eq 8). The values for k in Table 2 were plotted versus 1/T (where temperature is in Kelvin) as shown in Fig. 5. The [E.sub.a] of 141 kJ/mol, calculated via an application of a linear fit to this plot, agrees well with those reported for similar materials by Sastri et al. (8), Tan (9). Hinkley (14), and Takeshoshi and Terry (15).
1n[[kappa](T)] = 1nZ + (-[E.sub.a]/R) 1/T (8)
In addition to the [E.sub.a], the Arrhenius frequency factor, Z, may be determined from Eq 8 and the fit in Fig. 5; Z was determined to be 1.30 X [10.sup.9] 1/s. Knowing [E.sub.a] and Z for 2.5kPETU allows for the calculation of [[alpha].sub.I] at any time during an isothermal hold. This calculation utilizes Eq 9, which is obtained upon manipulation of Eq 7 and subsequent substitution of Eq 4. Figure 6 shows the percent error in predicting the time to reach selected values of [[alpha].sub.I] using first-order kinetics for temperatures ranging from 325[degrees]C to 360[degrees]C. These results demonstrate that first-order kinetics do not accurately predict [[alpha].sub.I] for isothermal cures in the temperature range of 325[degrees]C to 360[degrees]C.
[alpha] = 1 - exp [- Z t exp (-[E.sub.a]/RT)] (9)
Further investigation of first-order reaction kinetics includes analysis of the relationship between d[[alpha].sub.I]/dt and [[alpha].sub.I]. Equations 1 and 2 may be used to calculate d[[alpha].sub.I]/dt for any given [[alpha].sub.I] using first-order reaction kinetics. Figure 7 demonstrates the minimal correlation between the actual experimental d[[alpha].sub.I]/dt versus [[alpha].sub.I] curve and the calculated d[[alpha].sub.I]/dt versus [[alpha].sub.I] curve obtained with first-order reaction kinetics. The shape of the experimental d[[alpha].sub.I]/dt versus [[alpha].sub.I] curve appears to match better the combination cure kinetics represented by Eqs 5 and 6 than first-order kinetics represented by Eqs 1 and 2. These results along with the poor prediction ability shown in Fig. 6 call for an attempt to determine the necessary kinetic parameters to model the cure of 2.5kPETU via Eq 5.
Determination of the kinetic parameters in Wq 5 will be done by directly fitting Eq 5 to plots of isothermal reaction rate versus isothermal degree of cure. Close inspection of these plots reveals a change in slope between [[alpha].sub.I] 0.85 and [[alpha].sub.I] = 0.90. This slope change appears in most of the isothermal measurements near the same range of [[alpha].sub.I] and demonstrates an increase in the rate at which the reaction rate is declining. It is thought that this slope change indicates diffusion control is replacing chemical control as the rate controlling portion of the reaction. Since the cure kinetics models introduced earlier are suited for modeling only chemically controlled reactions and not diffusion controlled reactions, the combination cure kinetics model will be developed based on experimental data for: 0 < [[alpha].sub.I] < 0.90.
Curve fitting software utilizing the Levenburg-Marquardt algorithm determined the best values of [k.sub.1], [k.sub.2], m, and n for each isothermal temperature as shown in Fig. 8 for 340[degrees]C. The values determined from the curve fits are given in Table 3.
With values of [k.sub.1] and [k.sub.2] for each temperature, the Arrhenius frequency factors ([Z.sub.1] and [Z.sub.2]) and activation energies ([E.sub.a1] and [E.sub.a2]) of Eq 5 may be determined as [E.sub.a] and Z were determined previously for first-order kinetics. These parameters were determined by looking at [k.sub.1] and [k.sub.2] separately via Eq 8 as shown in Figs. 9 and 10. Knowing the values of [Z.sub.1], [E.sub.a1], [Z.sub.2], and [E.sub.a2] allows [k.sub.1] and [k.sub.2] to be calculated for any temperature within the experimental temperature range (Table 4). The values of m and n were plotted versus temperature (Figs. 11 and 12) to determine the temperature dependence of m and n allowing for the calculation of m and n at each experimental temperature (Table 4).
Enough information is known that at any temperature, d[[alpha].sub.I]/dt can be correlated with [a.sub.1] via Eq 5 using the temperature dependencies determined in Figs. 9 through 12. The difference between calculated d[[alpha].sub.I]/dt versus [[alpha].sub.I] plots and actual experimental d[[alpha].sub.I]/dt versus [[alpha].sub.I] plots was unsatisfactory. Through trial and error, n was found to have more of an influence than m on the quality of agreement between the calculated plot and the experimental plot, which would be expected from analysis of Eq 5. Thus the calculated d[[alpha].sub.I]/dt versus [[alpha].sub.I] plots were repeated using the temperature dependence of [k.sub.1] (Fig. 9 and Table 4), [k.sub.2] (Fig. 10 and Table 4), and m (Fig. 11 and Table 4) to determine these kinetic parameters. However, the values of n were taken from Table 3 rather than from Fig. 12 resulting in much improvement in the agreement between the calculated d[[alpha].sub.I]/dt versus a1 and the experimental d[[alpha].sub.I]/dt versus [[alpha].sub.I] relations.
The information garnered up to this point allows for realization of the objective of this kinetics modeling experiment; [[alpha].sub.I] versus time and temperature predictions were made by inserting the appropriate kinetic parameters ([k.sub.1], [k.sub.2], and m from Table 4 and n from Table 3) in Eq 6 for each temperature. Equation 6 was then rearranged and numerically integrated repeatedly to calculate the time to reach set values of [[alpha].sub.I]. A representative comparison of experimental and predicted [[alpha].sub.I] versus time relationships is shown in Fig. 13. Figure 14 and Table 5 show the percent error in predicting the time to reach selected values of a1 using combination kinetics for temperatures ranging from 330[degrees]C to 360[degrees]C. The results presented in Figs. 13 and 14 demonstrate that combination kinetics can be used to accurately predict [[alpha].sub.I] for isothermal cures in the temperature range of 330[degrees]C to 360[degrees]C.
Though the determination of kinetic parameters was based on [[alpha].sub.I] < 0.9, the fits are displayed up through [[alpha].sub.I] = 0.99 for the sake of comparing the abilities of first-order reaction kinetics and combination reaction kinetics to predict [[alpha].sub.I] versus time.
Cure kinetics models were developed utilizing both first-order reaction kinetics and combination reaction kinetics. The combination reaction kinetics model is significantly more accurate than the first-order reaction kinetics model (Figs. 6 and 14). Deviations between the models and experimental results at [alpha].sub.I] 0.9 can be attributed to what appears to be a transition from a chemically controlled reaction to a diffusion controlled reaction.
The empirical model that was developed based on combination kinetics is temperature specific in that the reaction order n is not given a fixed value or temperature dependence. Rather, n is individually selected for each temperature (Table 3). This selection of n significantly increases the agreement between predictions and experimental results. The combination reaction kinetics model works well for detailed prediction of degree of cure and reaction rate at any time throughout the whole range of cure for a selected temperature (Figs. 13 and 14, Table 5). If the model is made more general by giving n a fixed value or temperature dependence, accuracy is sacrificed. The first-order reaction kinetics model works better for quick estimation of degree of cure versus time for any temperature between 330[degrees]C and 360[degrees]C. The calculations can be done by hand and do not require numerical integration to calculate [[alpha].sub.I] at any given time. Furthermore, the first-order reaction kinetics model is not te mperature specific.
In the literature, there is a lack of consensus on the reaction order of PET mers. Some researchers (8, 9, 15) suggest that these reactions can be described by a simple order, whereas Hinkley (14) states that the reaction cannot be described by a simple order. The results presented here show that whether the reaction order can be represented by a simple order depends on the level of analysis. If one simply looks to determine [E.sub.a] and Z and predict [[alpha].sub.I] it appears as though the reaction may follow first-order kinetics. However, upon attempting to verify that the reaction fits the shape of a d[alpha]/dt versus curve for an nth-order reaction, one will find that the reaction does not appear to be nth-order (Fig. 7). Analysis via combination reaction kinetics produces two reaction orders, m and n, for each temperature. Figures 11 and 12 demonstrate that neither m nor n has a single value over the range of experimental temperatures, and though their values in general increase with increasing temper ature, the correlation coefficients for linear fits of m and n versus temperature are not high. In conclusion, the reaction of 2.5kPETU cannot be described by a simple order and further investigation is necessary to gain a better understanding of the reaction orders.
The cure kinetics models developed are based on isothermal DSC measurements. The chemically controlled portion of the reaction can be modeled well. However, it is apparent that diffusion control overtakes chemical control. Thus, the end of the isothermal cures measured by DSC are not fit well by the developed models. Furthermore, other studies (12, 13) have shown that cure of PET imide oligomers appears to continue despite being undetectable by DSC and FTIR. Therefore, additional information regarding cure was sought via other experimental techniques and methods. Discussion of these investigations may be found in reference 22.
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[FIGURE 14 OMITTED]
Table 1. Values of [DELTA][H.sub.I] and [t.sub.s] for Each Isothermal Cure Temperature. Temperature ([degrees]C) [DELTA][H.sub.I] (J/g) [t.sub.s] (min) 325 107.6 6.03 330 111.1 3.34 335 101.0 3.07 340 110.2 3.05 345 91.2 3.05 350 101.0 2.01 355 97.4 2.96 360 86.2 2.03 Table 2. Rate Constants Determined from Fits of Eq 7 to Experimental Data. Temperature ([degrees]C) End Time of Fit (min) 325 175.5 330 100.0 335 90.0 340 87.3 345 64.0 350 68.8 355 24.5 360 25.5 Temperature ([degrees]C) Corresponding [[alpha].sub.I] 325 0.995 330 0.998 335 0.999 340 0.996 345 0.999 350 0.996 355 0.946 360 0.999 Temperature ([degrees]C) Correlation Coefficient (R) 325 0.999 330 0.984 335 0.995 340 0.998 345 0.999 350 0.998 355 0.999 360 0.997 Temperature ([degrees]C) k (1/s) 325 5.15 x [10.sup.-4] [+ or -] 0.30 x [10.sup.-4] 330 1.00 x [10.sup.-3] [+ or -] 0.19 x [10.sup.-3] 335 1.23 x [10.sup.-3] [+ or -] 0.13 x [10.sup.-3] 340 1.06 x [10.sup.-3] [+ or -] 0.07 x [10.sup.-3] 345 1.79 x [10.sup.-3] [+ or -] 0.12 x [10.sup.-3] 350 1.30 x [10.sup.-3] [+ or -] 0.10 x [10.sup.-3] 355 2.06 x [10.sup.-3] [+ or -] 0.20 x [10.sup.-3] 360 4.12 x [10.sup.-3] [+ or -] 0.12 x [10.sup.-3] Table 3. Combination Cure Kinetics Model Parameters Determined From Curve Fits of d[[alpha].sub.I]/dt versus [[alpha].sub.I] for Each Temperature (Fig. 8). Temperature [k.sub.1] x [10.sup.-4] [k.sub.2] x [10.sup.-4] ([degrees]C) (1/s) (1/s) m 325 3.12 0.886 0.154 330 3.90 1.30 0.140 335 6.63 0.491 0.0603 340 7.42 2.16 0.258 345 10.6 4.03 0.361 350 13.1 3.57 0.276 355 18.2 7.43 0.610 360 27.5 4.26 0.317 Temperature ([degrees]C) n 325 0.835 330 0.722 335 0.775 340 0.836 345 0.928 350 1.07 355 1.09 360 0.895 Table 4. Combination Cure Kinetics Model Parameters Determined From Calculation Using the Fits of Figs. 19-12. Temperature [k.sub.1] X [10.sup.-4] [k.sub.2] X [10.sup.-4] ([degrees]C) (1/s) (1/s) m 325 3.07 0.724 0.096 330 4.21 1.01 0.15 335 5.75 1.39 0.20 340 7.81 1.91 0.25 345 10.6 2.61 0.30 350 14.2 3.55 0.35 355 19.0 4.81 0.40 360 25.3 6.48 0.45 Temperature ([degrees]C) n 325 0.76 330 0.80 335 0.84 340 0.87 345 0.91 350 0.95 355 0.99 360 1.03 Table 5. Percent Error in Calculated Time to Reach Predicted Degree of Cure (0.90 - 0.99) for Combination Kinetics (Fig. 14). Percent Error [alpha] 330[degrees]C 340[degrees]C 350[degrees]C 360[degrees]C 0.90 0.401 2.00 5.34 2.41 0.91 0.861 2.20 5.39 2.20 0.92 1.39 2.62 5.54 2.13 0.93 1.97 3.29 5.89 1.99 0.94 2.57 4.42 6.29 1.66 0.95 3.13 6.12 6.96 1.40 0.96 3.46 8.51 7.82 0.993 0.97 3.39 12.2 8.84 0.376 0.98 2.46 17.8 10.1 0.350 0.99 0.360 23.5 11.4 1.45
The authors would like to acknowledge the National Science Foundation Science and Technology Center for High Performance Polymeric Adhesives and Composites for support under grant DMR 91-20004. In addition, the Department of Defense National Defense Science and Engineering Graduate Research Fellowship Program's support of this research is appreciated. Also, financial support by the Virginia Space Grant Consortium is appreciated.
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|Author:||Bullions, Todd A.; Mcgrath, J.E.; Loos, A.C.|
|Publication:||Polymer Engineering and Science|
|Date:||May 1, 2002|
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