Modeling differential scanning calorimetry

Abstract

The present study is a unified mathematical approach to the analysis of DSC data. It addresses the baseline correction to the DSC record as well as the solutions of DSC curves of single or multiple, physical and chemical transformations. In this study, previous results in the pertinent literature become particular solutions of the general model. The transition baseline in a physical transformation is identical with the instrumental signal. When an ‘empty’ reference pan is used and the results are extrapolated to zero heating-rate, a pseudo-baseline can be defined. Physical transformations in DSC display a straight line signal during transition, followed by an exponential curve during the post-transition state. The model gives solutions for temperature and heat of transformation in both single and multiple physical transformations. The equation of the transition baseline in a chemical transformation accounts for: the pre-transition baseline; the heat-capacity change from that of the reactants to that of the products of the reaction; and the heating-rate difference between sample and reference materials. If there is no change in the heat capacities and either the reference pan is ‘empty’ or the thermal resistance is negligible, then the transition baseline is an extension of the pre-transition baseline. The DSC curve represents the corrected DSC record. The DSC curve of a chemical transformation is a bell-shaped graph whose skewness depends on the order of reaction. From the distinctive features of the DSC curve (i.e., relations at curve peak, inflection points and curve-bounded area), the apparent kinetic parameters are calculated (order of reaction, activation energy, pre-exponential factor and heat of reaction). The parameters specifically involved in these calculations are: peak temperature, inflection point temperature(s), shape index, area before peak, area after peak, asymmetry index, peak area and partial areas. The larger the asymmetry index, the smaller the reaction order. Final solutions contain the heating rate and the initial concentration (initial amount of reactant) as parameters. A zero-order chemical reaction displays an exponential signal and its solutions are particular. Two multiple reaction systems in DSC are analyzed: (a) two irreversible, first-order, parallel reactions, and (b) two irreversible, first-order, consecutive reactions. In the first case, the apparent kinetic parameters correlate with the area before the maxima/minima of the DSC curve. In the second case, the main parameter is the deflection at the maxima/minima. General-order, mechanistically uncoupled reactions are also discussed. Emphasis is placed on the assumptions and the analytical development as well as on the final solutions and their applicability range. The mathematics of physical transformations are referenced with time as an independent variable, whereas chemical transformations relate to temperature. No attempt was made to express the final solutions in both time and temperature coordinates.