# A Simple Direct Method to Obtain Kinetic Parameters for Polymer Thermal Decomposition
## 1 Introduction  
The effect of fire or heat and the associated fire hazards of the materials are studied through different phenomena. One of them is the thermal decomposition, which occurs when chemical compounds are heated. These thermal decomposition reactions are defined by the amount of gaseous fuel released, the release rate, and the temperature of each reaction. This phenomenon is fundamental to understand and simulate fire and its propagation between different elements. Therefore, a pyrolysis sub-model is a fundamental part of fire computational models [1]. Thermal analysis comprises a group of experimental techniques that measure changes in physical, mechanical, and chemical properties during controlled heating [2, 3]. The thermal decomposition characteristics of a solid may be determined by the kinetic analysis of the data [4, 5, 6]. Rates are parameterized in terms of temperature, pressure, and the degree of conversion, \(\alpha\), which is defined as:  
\[\alpha=\frac{m_{0}-m}{m_{0}-m_{f}} \tag{1}\]  
where \(m_{0}\) is the initial mass, \(m_{f}\) is the final mass, and \(m\) is the mass at a given time. \(\alpha\) is a dimensionless parameter. The influence of pressure is frequently ignored in kinetic studies, and most kinetic methods consider the rate to be a function only of temperature and the degree of conversion [7]. The rate constant, \(k(T)\), describes the velocity of the reaction as a function of temperature, and the reaction model, \(f(a)\), describes the dependence on \(\alpha\). The rate of thermal decomposition, \(da/dt\), is expressed as\[\frac{d\alpha}{dt}=k(T)\cdot f(\alpha) \tag{2}\]  
Although there are other approaches for representing \(k(T)\)[8], the Arrhenius expression, given by Equation (3)--where \(A\) is the pre-exponential factor (s\({}^{-1}\)), \(E\) (J mol\({}^{-1}\)) is the activation energy [2], \(R\) is the gas constant (J mol\({}^{-1}\) K\({}^{-1}\)), and \(T\) is the temperature (K)--is commonly used to describe the temperature dependence of a rate process [9]. \[k(T)=A\cdot e^{(-\frac{E}{k+T})} \tag{3}\]  
Although \(A\) and \(E\) are attributed to the frequency of molecular collisions and to the energy barrier of chemical bond breaking, respectively, as a mechanistic interpretation by some authors, these quantities may be considered to be adjustable parameters of \(k(T)\)[10]. The influence of \(\alpha\) on the rate process may be expressed in terms of accelerating, decelerating, or sigmoidal models. We focus on the decelerating model in this paper, because this formalism is commonly applied to thermal decomposition reactions in fire science [1; 2]. The decelerating model applies to processes where the rate initially exhibits its maximum value and decreases continuously as the degree of conversion increases. The most widely applied description of this process is the following reaction-order expression, which depends on \(\alpha\) and the reaction order, \(n\) (dimensionless parameter):  
\[f(\alpha)=(1-\alpha)^{n} \tag{4}\]  
The kinetic parameters, \(E\), \(A\), and \(n\), provide the ability to generate predictive thermal curves which can be used to assess the rate of thermal decompositions in terms of percent conversion, time, and temperature. These thermal decompositions can be of different nature, as melting crystallization, pyrolysis, oxidative reactions, etc. There are several methods for obtaining the kinetic parameters to describe the thermal decomposition of polymers [5; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26]. However, each method has particular constraints and limitations and may require some factors to be ignored to simplify interpretation [2]. The two approaches developed to study the kinetic parameters of thermal decomposition can be categorized as analytical and curve-fitting optimization methods [2] (see Figure 1). These are further divided into model-fitting and model-free methods. In model-fitting methods, the kinetic parameters are defined in terms of previously established reaction mechanisms.