Isothermal Crystallization Made Easy: a Simple Model and Modest Cooling Rates

An understanding of the kinetics of the crystallization process is important for the selection of processing parameters such as mold temperature and hold time during injection molding. Differential scanning calorimetry (DSC), is an excellent tool for following the progress of crystallization. A popular method for obtaining kinetics data known as isothermal crystallization, is based on rapidly cooling the sample from the melt to the crystallization temperature and then measuring the heat evolved while the sample is held isothermal. The model most often applied to isothermal crystallization data is the Avrami model. This model, while possessing physically significant parameters, is somewhat difficult to analyze. In this study, a simpler kinetic model, the Sestak-Berggren model, is applied to crystallization kinetics. The equivalency of the resultant kinetic parameters to those from the Avrami model is demonstrated. Furthermore, it is shown that modest cooling rates, as low as 5C/minute, may be used to obtain good kinetic results. INTRODUCTION AND THEORY In DSC, heat flow is measured as a function of time and temperature. When a material crystallizes, a measurable amount of heat is evolved, resulting in an exothermic peak in the DSC thermal curve. The shape of this peak is directly related to the kinetics (time and temperature dependency) of crystallization. Two fundamental properties which can be measured in a DSC are the reaction rate (dα/dt) which is related to the amount of heat flow at any given time and temperature, and conversion level (α) which is measured as the amount of heat evolved (enthalpy) from the beginning of reaction until the selected time and temperature. The mathematical equations used to model kinetic reactions generally take the form of dα/dt = k(T) f(α) where reaction rate is proportional to the specific rate constant (k) and is some function of the conversion level [f(α)]. The specific rate constant itself is a function of temperature (T), the dependence of which is often described by the Arrhenius equation, k = Z exp (E/RT). [Z is the pre-exponential factor, E the activation energy and R the universal gas constant]. A large number of kinetic methods have been developed which are based on different forms of f(α). For decomposition kinetics (evaluated by TGA) and reaction kinetics (measured by DSC), for example, f(α) is derived from the general rate equation and has the form: f(α) = (1-α)n (1) where n is the reaction order. This form of the kinetic equation works well for many simple, single-stage reactions. It is, however, inadequate when applied to more complex reactions such as auto-catalyzed chemical (e.g., thermoset cure) and polymer crystallization reactions. This is due to the multi-stage nature of these more complex reactions. Crystallization, for example, is a two-step process where crystal growth (step 2) takes place at nucleation sites whose appearance (step 1) is controlled by both time and temperature. The most popular form of f(α) for examination of auto-catalyzed reactions is known as the Sestak-Berggren (SB) equation [1,2]: f(α) = αm (1-α)n (2) where m and n are reaction order constants. The general rate equation (1) and the SB equation (2) are closely related. The general rate equation may be thought of as a simplified case of the SB equation where m is equal to zero. This is aesthetically satisfying to many practitioners since it has the appearance of reducing the number of applicable concepts which must be considered. For polymer crystallization, the most popular form of f(α) is: f(α) = r (1-α) [-1n (1-α)]1-1/r (3) where r is called the Avrami constant. This equation was independently derived by a series of authors [3-7], and is popularly known at the Avrami equation. Although the Avrami parameters are more difficult to evaluate than those of the general rate or SB equation, equation (3) is derived from first principles and hence, its parameter (r) has physical significance. The value of r depends on the shape of the nuclei and the dimensionality of their growth, as well as on the rate of their formation [8]. When nucleation sites are instantaneously formed, r has the value of 1 for needle-shaped crystals, 2 for plates, and 3 for spheres. If additional nucleation sites sporadically appear with time, the value for r is one integer higher. Most polymer crystals are anticipated to be spherical in nature, so values of r between 3 and 4 are most common. Because of its widespread availability to an easy-to-use data analysis software form, the SB equation is often used to model polymer crystallization processes with surprisingly good fit. The empirical observation that the SB equation may be applied to polymer crystallization, is supported in theory by the work of Sestak, Satava and Wendlandt who have shown that the SB equation is equivalent to the Avrami equation, to a first approximation [9]. Further, the relationship between the values of m, n and r may be obtained by equating α's from the two equations at the peak of the reaction curve where d(dα/dt)/dt = 0 [10]. This results in the relationship: r = 1 / [1 + 1n(n) 1n(n + m)] (4)