Data Analysis with the Vogel−Fulcher−Tammann−Hesse Equation

The Vogel−Fulcher−Tammann−Hesse (VFTH) equation has been the most widespread tool for describing the temperature dependence of the characteristic times of the cooperative dynamics of glass-forming liquids, both in low-molecular-weight and macromolecular systems near and above the glass transition temperature. Previous arguments have shown that statistical correlation could exist between the three parameters. The analysis of a representative experimental data set of poly(vinyl acetate) dielectric relaxation (Richert, R. Physica A 2000, 287, 26) allowed us to conclude that correlation exists mainly between the B and T0 parameters. It was found that the best fitting procedure should use a singular value decomposition implementation of the Levenberg−Marquardt nonlinear fitting algorithm. Simple criteria to diagnose (fitting matrix singular values) and minimize (linear combinations of fitting parameters responsible for instabilities edited to zero) ill-conditioned problems are given. The constant χ2 confidence limit ellipsoids are presented, and it was concluded that the expected association between fitted parameters should always be made with the partial correlation instead of the more familiar correlation matrix. Some qualitative arguments are presented, stressing the need to properly weight experimental data points whenever unequal statistical weights are present in actual data because of either the merging of data sets obtained from different methods or the transformation from the measured value into a different fitted variable. It is expected that the same trends will be evident to a greater or lesser degree for all instances of VFTH data analysis.