Autocatalytic curing kinetics of thermosetting polymers: A new model based on temperature dependent reaction orders

Abstract

Herewith we discuss a new model for thermoset polymers that follow the autocatalytic curing kinetics. This model is proposed upon investigation of the crosslinking reaction of 2,2′-Bis(4-cyanatophenyl)iso-propylidene (BACy), under isothermal conditions over a range of temperatures between 180 °C and 260 °C without catalyst. BACy undergoes crossslinking via a trimerization mechanism of the nitrile groups following an autocatalytic kinetics rather than an nth order kinetics. Comparing with other autocatalytic kinetics models, the new model takes into account that the reaction orders of the curing reactions in the polymers are temperature dependent variables rather than constants. The new model provides excellent agreement with the experimental data in a wide range of conversions and reaction temperatures.

Introduction

Thermosetting resins including cyanate esters or polycyanurates continue to play an important role in several key industries including aerospace, energy, automotive, coatings, microelectronics and optoelectronics because of their versatility in tailoring desired properties. Their applications span structural, such as those in composites, and functional, such as those in lithography, waveguides, and other photonic devices. Understanding and accurate controlling of the curing process is a pre-condition to select suitable processing parameters and to achieve optimum properties. It is, therefore, imperative to study the kinetics of the cure reactions, as it is a governing factor of morphology and structure development of the polymers. Many studies have been conducted on the curing kinetics of thermosetting polymers, and a variety of kinetic models have been proposed to relate the reaction rate to time, temperature, and conversion [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. In general, the kinetics models of thermosetting resins fall into two main types, (1) nth order model and (2) autocatalytic model because there are two categories of crosslink reactions differentiated by whether the maximum rate of reaction occurs at zero conversion or at a finite conversion value. The nth order model is often simple expressed by the reaction rate equation [1]:

$$\frac{d\alpha }{dt}=k{\left(1-\alpha \right)}^n$$

where α is the conversion, n is the reaction order, dα/dt is the reaction rate, and k is the apparent rate constant. Equation (1) automatically assumes that n is a constant independent of temperature and conversion and predicts a linear relationship between ln(dα/dt) and ln(1 − α). In most studies so far, these assumptions were accepted although there are often large discrepancies from the reality. In our earlier study, we attempted to address such discrepancies by recognizing that the assumed linear relationship is often only valid at very low conversion and substantial deviation was seen at moderate and high conversion. An alternative model was proposed which was able to provide excellent description of reactions that follows nth order kinetics [6].

However, the nth order model cannot describe the progress of complex reactions of some thermosetting resin systems in which autocatalysis presents [8], [9], [10], [11], [18], [19], [20], [21], [22], [23], [24], [25]. For example, Equation (1) predicts the maximum of reaction rate at zero conversion which is not the case for the category of thermosetting resins including epoxy/amine systems and some bismaleimides [11] and the polycyanurate reported in this study. Instead there is a different model proposed first by Kamal et al. [18], [19] based on studies of epoxy resins which represents autocatalytic curing kinetics as shown in Equation (2).

$$\frac{d\alpha }{dt}=k_1{\left(1-\alpha \right)}^n+k_2{\alpha }^m{\left(1-\alpha \right)}^n$$

where k₁ and k₂ are two individual rate constants following Arrhenius temperature dependency. Two reaction orders m and n are introduced that are constants. The autocatalytic kinetics model derived from Equation (2) predicts maximum reaction rate (dα/dt) at an intermediate conversion (α > 0). Mathematically, at the point of maximum reaction rate, d(dα/dt)/dα = 0, hence the following equation can be derived from Equation (2):

$$\left({\alpha }^m-\frac{m}{m+n}{\alpha }^{m-1}\right){|}_{\frac{d\alpha }{dt}={\left(\frac{d\alpha }{dt}\right)}_{\text{max}}}=-\frac{n}{m+n}\frac{k_1}{k_2}$$

Usually k₁ << k₂, then we have:

$${\alpha }_{\max }=\alpha {|}_{\frac{d\alpha }{dt}={\left(\frac{d\alpha }{dt}\right)}_{\text{max}}}\approx \frac{m}{m+n}$$

For a typical autocatalytic reaction, α_max, the conversion at which the reaction rate reaches its maximum is around 0.3–0.4 [26]. This model has been found useful and received much attention [8], [9], [10], [11], [18], [19], [20], [21], [22], [23], [24], [25] because it is able to address the inability of the Equation (1) in describing reactions of many thermosetting polymers when maximum rate of reaction is not at zero conversion (t = 0 or α = 0) due to autocatalysis. However, controversy remains in literatures because of the limitations of this model. One of the limitations arose from the fact that both m and n are not necessarily constants, although this model is still widely used even though serious inconsistency exists between the theory and experimental data because of lack of alternative models. This work is therefore devoted to derive a new kinetics model for autocatalytic curing reactions. Curing kinetics of catalyzed cyanate ester systems has been reported and different techniques, such as DSC [27], [28], FTIR [29], and DEA (Dielectric analysis) [30] have been employed to monitor the curing reaction. Polycyanurate is selected in this study not only because of its proven significance in aerospace and electronics [31], [32], [33], [34], [35] and its potentials in optical telecommunications, [36], [37], [38], [39], [40], [41], [42] but also because it serves as a good model compound for reaction kinetics studies. Its reaction may follow either nth order or autocatalytic kinetics depending on whether a catalyst is used. However, very few studies were reported on the curing kinetics of uncatalyzed aromatic polycyanurates [43], [44]. Although the catalyzed polycyanurate systems are often used in composites applications, uncatalyzed systems are important for dielectric or photonic applications, because the use of catalysts may be detrimental to their dielectric or optical properties. In the present study, we selected 2,2′-Bis(4-cyanatophenyl)iso-propylidene (BACy) with no added catalyst, which provides an ideal platform to derive new autocatalytic kinetics models. By introducing a temperature dependency of reaction orders, an alternative model is derived which provides accurate prediction of autocatalytic reaction kinetics in thermosets.