Deconvolution of superimposed DTA/DSC peaks using the simplex algorithm
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Mathematical models emulating DSC/DTA traces of fusion and first-order decomposition were developed. A computer program is described that uses the simplex numerical optimization algorithm to determine the coefficients for the model equations that best fit experimental data, by the criterion of least-squared error. Use of the program permitted deconvolution of various superimposed endotherms.
I. INTRODUCTION Historically, thermal analysis techniques such as differential thermal analysis (DTA) and thermogravimetric analysis (TGA), used for the identification of raw materials such as clays, have lost favor with the development of x-ray diffraction and electron microscopy techniques. However, thermoanalytical techniques represent the only rapid method of determining the kinetics and thermodynamics1"7 of transformations as a function of time and temperature. With slower heating rates, transformation peaks in DTA and DSC (differential scanning calorimetry8) often show features implying the superposition of multiple peaks. If these individual peaks could be isolated, much information about the onset temperatures, rates, and mutual interdependence of individual reactions governing an overall transformation could be discerned. The concept of additive heat flow (via contributions from multiple transformation sources within the sample material) as measured by power-compensated DSC is reasonable—just as the water flow into one tank from two pipes would be additive. Assuming Fourier's law holds, e.g., steady-state heat flow proportional to temperature gradient, the temperature differences measured in DTA (and heat-flux DSC) may be additive. The process of "deconvolution" of superimposed endotherms/exotherms requires modeling individual reactions to heat-flow functions, which when added together, emulate the experimental data. The models derived herein, melting and first-order decomposition, are used as examples, and certainly do not exhaust the possibilities of transformation phenomena studied by thermal analysis. Rather than deconvolute experimental data, DSC/DTA data were "fabricated" by generating equations. In that way, the solution coefficients were known in advance, so that the capabilities of the deconvolution technique could be properly evaluated. In addition to deconvolution, the computer fitting of model equations to individual transformation peaks J. Mater. Res., Vol. 8, No. 3, Mar 1993 http://journals.cambridge.org
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has the utility of establishing important parameters of the reaction, such as reaction mechanism and activation energy. This sort of modeling has previously been undertaken by sometimes cumbersome and questionable9"11 mathematical manipulation of experimental data. Before actual data can be fit to a model, extraneous effects manifested in the trace must be removed, such as the shift in baseline as a result of the sample being different in nature, hence having different heat capacity, at the termination of a transformation as compared to the onset.8 It may, for some device design
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