Development of kinetics sub-model of cyanate ester-based prepregs for autoclave molding process simulation

Introduction

A prepreg lamina consists of a reinforcement material pre-impregnated with a resin matrix in controlled quantities. The resin is partially cured to B-stage for easy handling and is cured completely during the processing of the prepreg to make end products. Prepreg autoclave molding process has several advantages over traditional fiber-reinforced plastic (FRP) manufacturing processes. The advantages includes lower void content, control of fiber volume fraction and laminate thickness, clean process and lower labor cost. Due to these advantages, prepreg autoclave molding process is the choice of manufacturing process for high end products such as components of airplanes, missiles, high speed trains and fast ships. In manufacturing processes, prepreg laminate can be consolidated by applying pressure and curing reaction can be initiated and maintained by applying heat. Uneven curing inside the laminate which may affect the uniformity of the laminate thickness can be avoided by maintaining heating rate. So process modeling can be a very effective tool for these manufacturing processes and kinetic model is an important sub-model required for the process modeling.

Cyanate ester resin has a very stable performance at very high temperatures for its high glass transition temperature which is around 463 K with excellent electrical properties and good handling characteristics. So cyanate ester-based prepregs have a very wide application in aerospace and electrical industry. Various literature [1, 2, 3, 4, 5, 6] on the reaction mechanism showed that curing of the cyanate ester resin forms polycyanurate by cyclotrimerization of -O-C ≡ N functionality. It is very easy to find the degree of cure of a crosslinking reaction utilizing the exothermic heat of reaction evolved during cure but very difficult to find the mechanistic model for curing reaction, as complete mechanism of curing and the product composition of most of the polymeric crosslinking reactions is not accurately known. The degree of cure of a thermosetting resin is related to the heat of reaction by the following expression

where H(t) is the heat evolved as a function of time and H_r is the total heat of reaction.

The cure kinetics modeling involves measuring the parameters associated with the exothermic cure reaction and then developing a mathematical model for the cure rate as a function of temperature and degree of cure. A general expression for curing rate can be represented by eq. (2).

where dα/dt is the rate of reaction, k(T) is the temperature-dependent rate constant, and f(∝) corresponds to the reaction model. The temperature dependence of rate constant, k(T), is defined by Arrhenius equation.

From literature [4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] it was found that the reaction kinetics of thermosetting resins can be described either by nth order, presented by eq. (4), or by autocatalytic reaction presented by eq. (5), or the combination of both [18].

A modification to the autocatalytic kinetic model for the curing reactions where the initial cure rate is not zero was proposed by Kamal et al. [18] given by:

where k₁ and k₂ are the rate constants, α is the degree of cure, and m and n are the orders of reaction.

In many curing reactions the final values of α is not a unity and to incorporate that, eq. (6) is modified later to eq. (7)

where α_f is the final degree of cure.

Kamal’s autocatalytic model fails to predict the behavior at the later stages of the curing reaction which is diffusion controlled due to the formation of gel. The formation of gel offers resistance to the mobility of the reacting molecules. A number of investigators [2, 14, 19] have introduced a rate constant for the diffusion controlled region as,

where C and α_c are two empirical constants which are temperature dependent. α_c is called the critical degree of cure where the curing mechanism is converted to diffusion controlled reaction from a chemical controlled reaction. Some researchers [14, 19] modified the Kamal’s model given in eq. (9) by introducing a diffusion factor. Chern and Poehlein [20] came up with a model which is given by,

where f₁(α) is the diffusion factor and is expressed as:

Most of the published cure kinetic models of thermoset resin are related to the epoxy [11, 19], phenolic [9, 10, 12, 13, 21, 22], or unsaturated polyester resin [15]. However, hardly any literature is available related to the modeling of cure kinetics of formulated cyanate ester resin-based prepregs. Ching-Chung Chen et al. [4] have found that cure kinetics of cyanate ester resin can be described by autocatalytic model. Siddiqui et al. [16] have found the curing of a cyanate ester resin system as a multistep reaction scheme and modeled the kinetics independently for each cure zone and combined together to find the complete curing reaction. Not much literature is available for modeling the complex reactions in which multiple reactions are involved. J. M. Kenny et al. [13] have found that in the curing of phenolic resin, two separate reactions are present and they modeled the kinetics of overall curing by modeling the reactions separately and combined using a weighting factor.

In the present work, kinetic characterization of cyanate ester-based resin impregnated prepreg systems specifically formulated for autoclave molding process was performed and a model was developed. The complete work plan is presented in Figure 1. Exothermic heat evolved during curing process was captured by differential scanning calorimeter (DSC) in dynamic and isothermal mode. In dynamic mode, temperature of the DSC cells was varied with a constant ramp rate, as low as 0.5 K/min to as high as 10 K/min. Heat flow from the dynamic mode of DSC indicated that multiple reactions were involved in the whole curing process. As practiced by several researchers [7, 10, 12, 22, 23, 24] dynamic DSC data was used to find the activation energy of the overall reaction by Friedman’s model-free iso-conversional analysis [25] and to find the activation energies of individual reactions by the Kissinger method [26]. Finally, a detailed kinetic model to describe the cure rate as function of temperature and conversion was developed using the isothermal DSC data.

Figure 1:

Work plan.

Experimental

Materials

Cyanate ester resin-based quartz fabric prepreg has been used for kinetic characterization. The resin content of the prepreg was about 41 % (w/w) and the thickness of prepreg sheet was 0.34 mm. Exact formulation of the resin in the prepreg was unknown. However, FTIR spectroscopy of the resin extracted from the prepreg was performed to establish the chemical footprint of the resin used for future reference. The FTIR spectroscopy is presented in Figure 2. The chromatogram in Figure 2 confirms the presence of C ≡ N stretch (2220–2260 cm⁻¹), Ester C=O stretch (1735–1750 cm⁻¹), Carboxylic O-H stretch (2500–3000 cm⁻¹), Alcohol/Phenol O-H stretch (1630–1780 cm⁻¹), Alkenyl C=C stretch (1620–1680 cm⁻¹), Aromatic C=C bending (1500–1700 cm⁻¹). Because of the non-uniformity of the fiber content in the prepreg sample, it is very difficult to find the heat of reaction values of the resin. To avoid this problem, resin was first extracted from prepreg and the experiments were done on extracted resin from prepreg which is practiced by Siddiqui et al [16]. The prepreg was heated to 353 K and the resin was extracted with the help of a spatula. The temperature was selected as 353 K because at that temperature the resin has minimum viscosity which is about 0.3 Pa.S and it is lower than the curing onset temperature. Dynamic scans for both prepreg and extracted resin, presented in Figure 3(a) for a heating rate of 5 K/min, were performed to ensure that resin extraction process did not affect the kinetic parameters of resin.

Figure 3:

Dynalmic DSC scan: (a) Heat flow from prepreg and extracted resin at same heating rate (b) Normalized heat flow from extracted resin at different heating rate.

Figure 2:

FTIR spectroscopy scan of the extracted resin from the prepreg.

Results and discussion

The normalized heat flow vs. temperature at different heating rates in case of dynamic DSC is graphically represented in Figure 3(b). The normalized heat flow vs. time data obtained from the dynamic DSC experiments were converted into degree of cure (α) and rate of cure (dα/dt) using eq. (1). The change in the degree of conversion and the rate of conversion with the temperature are graphically presented in Figure 4(a) and Figure 4(b) respectively.

Figure 4:

(a) Degree of conversion and (b) rate of conversion in dynamic curing process at different heating rate.

Presence of two peaks in Figure 3(b) gives an idea about the complexity of the curing reaction. The whole curing reaction consists of two reactions with different activation energy values. Because of significant difference in activation energy values, two peaks are clearly visible for higher heating rates. Kissinger method [26] was applied to find the activation energies of these two reactions considering both follow nth order kinetics. Activation energies are found to be 73 kJ/molK and 82.3 kJ/molK. Model-free iso-conversional method (multiple heating rate method) [25] was applied to the dynamic conversion data assuming activation energy depends only upon degree of conversion and does not depend upon temperature. The activation energy as a function of degree of conversion is plotted in Figure 5 which gives an idea about the consistency of the curing mechanism throughout the curing process. Although the variation of activation energy with conversion was not so high, initially it has an increasing trend with conversion implying that the reaction turns to follow any other kinetics when some product is also present with the reactant and later it decreases due to the high product concentration. This also indicates that the curing reaction follows autocatalytic reaction kinetics at the later stages of the curing process.

Figure 5:

Change of Activation energy of curing at different degree of conversion.

Normalized heat flow with time obtained from DSC scan at isothermal temperatures of 398, 403, 408, 413, 418 and 423 K are presented in Figure 6. Heat evolved with respect to time was converted to degree of conversion using eq. (1) and is presented in Figure 7(a). Rate of conversion was calculated by differentiating degree of conversion vs time curve and is plotted against degree of conversion in Figure 7(b). For isothermal measurement, there were always errors involved due to the heat absorption of the sample to attain the isothermal temperature. To minimize this error, sample was put in the DSC cell after equilibrating the cell at 353 K which is below onset temperature and then heating the cell very rapidly (80 K/min) to isothermal temperature.

Figure 7:

(a) Degree of conversion and (b) rate of conversion in isothermal curing process at different temperatures.

Figure 6:

Normalized heat flow from isothermal DSC scan of extracted resin at different temperatures.

Two distinct zones of rate of reaction are clearly visible in Figure 7(b). The initial maximum rate of reaction implies that the reaction follows nth order kinetics which dominates the process up to a certain conversion value of 0.25 and beyond the conversion of 0.5, the shape of the curve is like an autocatalytic reaction where initial rate of reaction is not zero. So the complete curing reaction consists of an nth order reaction and an autocatalytic reaction.

Proposed kinetic model

The proposed kinetic model is a combination of an nth order model and an autocatalytic model which are connected by a weighting factor, q. The model is

(11)dαdt=qk11−αn1+1−qk2+k3αm1−αn2

where dα/dt is the rate of reaction; n₁, n₂ and m are the model parameters; k₁, k₂ and k₃ are the rate constants which follows Arrhenius equation.

The first term in the model dominates when the reaction follows nth order kinetics, i. e. at the initial stages of curing and the second term dominates when the reaction follows autocatalytic reaction kinetics, i. e. when some product is already produced. The model parameters are found by fitting the rate of conversion vs degree of conversion data for constant temperatures. Because of the large number of parameters present in the model, direct fitting of the data does not converge to a particular value. So the model was fitted in several steps and some parameter values were fixed in intermediate steps. In the first step of data fitting, when all the parameters were allowed to vary to fit the experimental rate of conversion data, the weighting factor was coming to almost constant for all the fits. So the weighting factor does not vary with temperature and degree of conversion. Hence, the weighting factor can be incorporated with the frequency factors of the rate constants. Then the simplified model becomes the summation of an nth order model and an autocatalytic model:

(12)dαdt=k11−αn1+k2+k3αm1−αn2

This model was fitted with the experimental data at different isothermal temperatures. Kinetic parameters n₁, n₂ and m for different temperatures were found to be very less variation and so, they were fixed by taking average. The experimental data was fitted again to find the corrected rate constant values after fixing all the temperature independent parameters. The final model parameters for different temperatures are listed in Table 1.

Table 1:

Model parameters at different temperatures.

T (K)	ko	k1	k2	n1	n2	m	R2
398	1.03E-3	1.27E-04	2.14E-3	16.83	1.59	7.225	0.999
403	1.09E-3	1.67E-04	2.62E-3	16.83	1.59	7.225	0.997
408	1.44E-3	2.27E-04	3.35E-3	16.83	1.59	7.225	0.975
413	1.53E-3	3.27E-04	4.92E-3	16.83	1.59	7.225	0.987
418	1.56E-3	4.55E-04	6.38E-3	16.83	1.59	7.225	0.998
423	1.69E-3	5.74E-04	7.97E-3	16.83	1.59	7.225	0.997

Activation energy and frequency factors were calculated from the Arrhenius plot shown in Figure 8 and the values are tabulated in Table 2.

Figure 8:

Arrhenius plot of the rate constants.

Table 2:

Parameters of rate constants from Arrhenius plot.

Rate constant	Activation energy (J/mol)	Frequency factor
k1	29,004.66	6.73273
k2	87,130.08	3.37E+07
k3	76,980.24	2.58E+07

The model was validated by predicting the reaction rate and comparing the reaction rate obtained from experiments shown in Figure 9(a). Validation of the model was also confirmed by comparing degree of conversion which is obtained by integrating the reaction rate predicted by the model numerically and the experimental degree of conversion. The generated degree of conversion along with the experimentally found degree of conversion against time is plotted in Figure 9(b). As the temperature range was selected certainly below the temperature at peak heat flow, the model needs to be validated at or above the peak heat flow temperature. For this purpose additional isothermal experiments were carried out in the temperature range of 463K–483K. The model simulated reaction rate and degree of conversion are plotted along with respective experimental data at higher temperature range which are presented in Figure 9(c) and Figure 9(d) respectively. Figure 9(c) clearly indicates that the initial conversion range, where the nth order reaction was assumed to be prevailed, cannot be evaluated as the reaction was already completed before attainment of the isothermal temperature. Figure 9(c) and Figure 9(d) proves that the proposed kinetic model is valid for lower temperature range as well as for higher temperature range.

Figure 9:

Comparison of experimental and model simulated (a) reaction rate at lower temperature range (b) degree of conversion at lower temperature range (c) reaction rate at higher temperature range (d) degree of conversion at higher temperature range.

Conclusions

A simple empirical kinetic model for cyanate ester-based resin has been developed which can be used as sub-model for autoclave process model. The methodology utilizes the exothermic heat evolved during the crosslinking reaction. The complexity of the curing reaction is given by dynamic DSC and the model-free kinetics gives the idea about the consistency of the reaction mechanism which is proved by isothermal DSC. The overall curing reaction follows nth order kinetics initially and after some conversion value, reaction scheme turns into an autocatalytic reaction but both the reactions were present to some extent throughout the curing. The overall kinetics has been modeled as a combination of nth order and autocatalytic reaction kinetics which can predict the rate of curing at all temperatures.