The role of an objective function in the mathematical modelling of wide-angle X-ray diffraction curves of semi-crystalline polymers

To decompose a wide-angle X-ray diffraction (WAXD) curve of a semi-crystalline polymer into crystalline peaks and amorphous halos, a theoretical best-fitted curve, i.e. a mathematical model, is constructed. In fitting the theoretical curve to the experimental one, various functions can be used to quantify and minimize the deviations between the curves. The analyses and calculations performed in this work have proved that the quality of the model, its parameters and consequently the information on the structure of the investigated polymer are considerably dependent on the shape of an objective function. It is shown that the best models are obtained employing the least-squares method in which the sum of squared absolute errors is minimized. On the other hand, the methods in which the objective functions are based on the relative errors do not give a good fit and should not be used. The comparison and evaluation were performed using WAXD curves of seven polymers: isotactic polypropyl­ene, polyvinylidene fluoride, cellulose I, cellulose II, polyethyl­ene, polyethyl­ene terephthalate and polyamide 6. The methods were compared and evaluated using statistical tests and measures of the quality of fitting.