«CURE KINETICS OF WOOD PHENOL-FORMALDEHYDE SYSTEMS By JINWU WANG A dissertation submitted in partial fulfillment of the requirements for the degree of ...»
The face resin displayed a weight-average molecular weight (Mw) of 621 g/mol and a polydispersity (Mw/Mn) of 1.4. This resin was subsequently identified as PF-low. The core resin possessed a Mw = 6576 g/mol and Mw/Mn =1.72 and was labeled as PF-high. Resin solids contents for the PF-low and PF-high resins were 54.5% and 45.0%, respectively. In addition, elemental analysis (Nelson and Sommers 1982) showed the presence of 3.7 and 3.9 wt % nitrogen for PF-high and PF-low respectively, suggesting the presence of urea in both resins.
Differential Scanning Calorimetry A Mettler-Toledo DSC 822e was used to perform dynamic and isothermal cure experiments. Approximately 13.5mg of resin was placed in a 30 µl high pressure gold-plated crucible. Dynamic temperature scans were conducted from 25°C to 250°C at 4 heating rates: 2, 5, 10 and 20°C/min. In all DSC scans, nitrogen was used as a purge gas at a flow rate of 80 ml/min. Six replicate measurements were performed for each heating rate. Four randomly selected measurements were used to extract kinetic parameters and remaining two measurements were used to compare with predictions.
In addition, the first replicate was re-scanned at 10oC/min immediately following the first scan to assure complete cure.
Both degree of cure (α) and reaction rate (dα/dt) were determined at a specific cure time (t) by normalizing the partial heat of reaction (∆H(t)), and heat flow (dH/dt) by the total heat of reaction (∆H), respectively:
The cure kinetic parameters for the nth order with Borchardt-Daniels, autocatalytic model with Borchardt-Daniels, ASTM E698 and modified autocatalytic methods were extracted form the cure and cure rate data using linear least-squares fitting routines programmed in MATLAB. The resulting kinetic parameters were then used to predict and compare dynamic cure with experimental data at 4 different heating rates. To further validate the methods for isothermal cure, isothermal DSC runs were conducted at 120°C for different times. A cure temperature of 120°C was representative of PF cure under typical hot-pressing conditions for the panel core (Zombori et al. 2004) and it also allowed easy observation of cure development with DSC. The DSC cell was preheated to 120°C and approximately 13.5 mg of PF resin was inserted and cured for different times. The sample was then quickly removed from the DSC and quenched in liquid nitrogen. The residual heat of reaction of the partially cured samples (∆HR) was obtained from a subsequent ramp scan at 10°C/min from 25 to 250oC. The time dependence of the degree of cure at 120 °C was obtained by normalizing the difference of total and residual heat of reaction with total heat of reaction (α=(∆H-∆HR)/∆H) as a function of cure time. The total heat of reaction was taken as the average reaction heat previously measured in dynamic tests of fresh resins. The time dependence of the degree of cure at 120°C was compared with predictions from the MF methods.
During a reaction process, the overall reaction rate can be modeled as:
where t (s) is the time, A (s-1) the pre-exponential factor, E (J/mol) the activation energy, R (8.314 J/mol·K) the universal gas constant, T(K) the absolute temperature and f(α) the reaction model. Two reaction models are commonly used for simple
f (α ) = α m (1 − α ) n in which m+ n is called the order of reaction. Under isothermal conditions, in nth order kinetics, the rate of conversion is proportional to the concentration of unreacted material. Reaction rate therefore reaches its maximum at the onset of reaction and then decreases until the reaction is complete (ASTM E2041).
In autocatalyzed kinetics on the other hand, both the reactant and product are catalysts so that a maximum reaction rate is obtained during the course of the reaction (Prime 1997). Both models can be applied to dynamic experiments (Martin and salla 1992).
Using the nth order and autocatalytic models Eq. (8) can be rearranged into Eq.
(4) and (5) respectively:
From one DSC dynamic scan, the values of α and dα/dt and corresponding temperature are used to solve Eqs. (9) and (10) by multiple linear regression. Kinetic parameters, A, E and n for the nth order model (nth-BD) and A, m, n, and E for the autocatalytic model (Auto-BD) are thus determined (ASTM E2041; Borchardt and Daniels 1957). This DSC analysis is usually designated as the Borchardt-Daniels method or the single heating rate method; it is attractive because all the kinetic parameters are derived from one single dynamic DSC scan. With this method however, kinetics parameters are heating rate dependent and they are subject to signal noise, solvent effect and unresolved baselines (Dunne et al. 2000).
The kinetic parameters can also be determined from multiple heating rate scans. Specifically, in the case of a constant heating rate, β=dT/dt, Eq. (8) can be
rearranged into the integral form g(α):
Because Eq. (11) has no exact analytical solution, Doyle (Doyle 1961; 1962) proposed two approximations for g(α), which can be rearranged into (Ozawa 1965;
Flynn and Wall 1966):
The peak temperature (Tipeak) dependency on heating rate (βi) can thus be used to calculate the activation energy (Kissinger 1956; Shulman and Lochte 1968; Wang et al. 1995). Assuming an iso-fractional peak temperature (Horowitz and Metzger 1963), a linear regression of ln (βi/T2ipeak) or log (βi) against 1/Tipeak across several heating rates yields the activation energy with Kissinger Eq. (14) (Kissinger 1957) or Ozawa Eq. (15) (Ozawa 1965; Shulman and Lochte 1968) respectively.
Ozawa equation yields slightly higher E values than those obtained by Kissinger equation (ASTM E 698; Alonso et al. 2004). The calculated activation energy by Ozawa equation can be refined as suggested by ASTM E698 to be comparable with that by Kissinger equation. This modified version of Ozawa equation is designated as E698 method. For PF resins, Alonso et al (2004) found less than 4% variation in the estimates of E between the two equations. In addition, the estimates of E by E698 method are lower than those from the Borchardt-Daniels method (Alonso et al. 2004;
Park et al. 1999). To calculate the pre-exponential factor A, a definite reaction model
must be assumed. For nth order reactions, Kissinger (1957) proposed:
Assuming a first order reaction, A is easily obtained from Eq. (16) by substituting the intermediate heating rate and its corresponding peak temperature. Hence, with the E698 method, E can be determined regardless of the reaction model while A can only be measured for nth order reactions.
Another method has been developed to calculate autocatalytic kinetic parameters from multiple heating rate DSC scans (Harper et al. 2001; Lam 1987;
Chung 1984). This method assumes that the degree of cure at the exothermic peak (αpeak) is heating rate-dependent and relates to the reaction orders m and n. The relationship between degree of cure at peaks and reaction orders is given by setting
the optimum criteria in Eq.(10):
For any specific kinetic process, αpeak is obtained experimentally, thus constraining the values of m and n by Eq. (17). Another constraint arises from the integral function
for the autocatalytic model in Eq. (18):
In this case, the zero value of lower limit of integral imposes a constraint: m must be less than unity for g(α) to be finite (Martin and Salla 1992). It is possible to obtain a unique analytical solution for g(α) when (n + m) sums to an integer higher than 1 and when m 1 (Martin and Salla 1992). However, for n + m = 1, the kinetic integral g(α) has different solutions for each value of n with m 1 (Martin and Salla 1992). In this paper, g(αpeak) is not solved analytically but rather numerically by assuming a value for the reaction order m + n of 1, 2 and 3 respectively within the constraints of Eq. (17) and that of m1. Activation energy and pre-exponential factor are then determined simultaneously from the slope and intercept of the log (βi)+log [g(αipeak)] versus 1/Tipeak plot across several heating rates βi according to Eq. ( 13). This method designated as the modified autocatalytic model (M-Auto) is advantageous in that the peak temperature is not assumed iso-fractional (Harper et al. 2001). Table 3.1 summarizes the mathematical expressions, parameters and algorithms associated with each of the models used in this paper.
Table 3.1 Summary of kinetic models, parameters and methods used.
RESULTS AND DISCUSSIONPF Cure analysis The commercial PF resins exhibit two distinct exotherms that shift to higher temperatures with increasing heating rate (Figure 3.1). PF-high displays an additional exothermic shoulder between the two major exotherms (Figure 3.1). In PF-high, the highest molecular weight fractions react rapidly so that the maximum exotherm appears early for PF-high (Detlefsen 2002). The PF-high resin reaches similar degree of cure ca. 10°C earlier than that required for the PF-low resin (Figure 3.1). As expected, DSC appears sensitive to differences in reaction exotherms and cure development of different resins.
Table 3.2 Summary of PF cure peak temperature, degree of cure at peaks in parenthesis and heat of reaction across 4 heating rates*.
Figure 3.1 (a) DSC heat flow (dH/dt) at 2 oC/min and (b) degree of cure (α) for PF-low and PF-high.
The number 1 and 2 designate exotherm peak 1 and peak 2.
Insert in (a) highlights the influence of heating rate (2, 5, 10, and 20 oC/min) on the cure of PF-high.
rates for each resin. As expected the more advanced PF-high resin releases less heat corresponding to the cure reaction (365 ±5 kJ/kg) than does the PF-low resin (420 ± 9 kJ/kg) (Table 3.1). In addition, the peak temperatures were found to be approximately iso-fractional regardless of heating rate (Table 3.2).
Table 3.3 Kinetic parameters for the PF-low and PF-high Resins obtained from the model-fitting kinetic methods.
a: average of 4 replicates at each heating rate, standard deviation ≤ 2.90 Kinetic parameters from model-fitting kinetics Table 3.3 summarizes the kinetic parameters obtained by the MF kinetics from dynamic test data for the two PF resins. The Auto-BD leads abnormal kinetic parameters and is therefore not applicable for PF resins. The Auto-BD method is unable to account for all intrinsic properties of the autocatalytic model, i.e. the constraints of m and n (Eqs. (17) and (18)) are not met with this method.
The nth-BD, E698 and M-Auto methods generate consistent activation energies and pre-exponential factors for both resins (Table 3.3). These parameters are in the 83-105 kJ/mol and 19-26 s-1 ranges respectively and are in agreement with the literature (Kay and Westwood 1975) and with model-free kinetics methods (Wang et al. 2005). With all three methods, slightly higher activation energy is found for PF-high than that for PF-low, consistent with the higher advancement of PF-high (Vazquez et al. 2002).
The advantage of the E698 and M-Auto methods over the nth-BD method is that kinetic parameters can be determined for individual exotherm. The M-Auto method is only applicable when a reaction order (m) is small. When the M-Auto method can be applied, it generates activation energies slightly lower than those measured with the E698 method (Table 3.3). Recall that the E698 method neglects the dependence of αpeak on heating rates whereas this dependence is included in the M-Auto method (Harper et al. 2001). For the two PF resins used in this study, there are only small variations for degree of cure at peaks across heating rates. As a result, both methods lead small differences in activation energies for the two resins. The nth-BD method gives cure kinetic parameters for the overall cure process at each heating rate, which are very similar to those obtained from maximum peaks with E698. A trend towards higher activation energies with increasing heating rates for the PF-low resins is observed (Table 3.3). The higher activation energies measured at 20 oC/min suggests that this heating rate is less appropriate to characterize PF cure. This discrepancy is likely due to the higher thermal lag manifested at the higher heating rate. The nth-BD method is generally observed to overestimate the kinetic parameters when compared with the E 698 method (Alonso et al. 2004; Park et al. 1999). However, with the exception of the 20 oC/min, this overestimate is not apparent in this study. It is clear that the nth-BD and E698 methods can provide consistent kinetic parameters, and the M-Auto method is limited to small reaction orders while the Auto-BD method is inapplicable.
Predicting cure for dynamic conditions Dynamic cure development was predicted by substituting values of activation energy, pre-exponential factor, reaction orders, and arbitrary heating rates for the corresponding nth order model or autacatalytic model into Eq. (8). The equations were then solved using the Runge-Kutta method (Dormand and Prince 1980) implemented in a MATLAB program. All kinetic parameters from Table 3.3 can be used to predict cure behavior. Further, with the E698 method two reactions can be modeled as independent and/ or consecutive reactions (Eq. 19) (Flynn and Wall 1966). Two reactions can also be modeled as parallel and competing (Eq. 20) (Vyazovkin 2001).
In both cases, the overall reaction rate is obtained by substituting the kinetic
0.4 0.4 0.2 0.2
Figure 3.2 Comparison of the test data at 10 oC/min and MF predictions of PF-low and PF-high for reaction rate (dα/dt) and degree of cure (α).