The amorphous state: first-principles derivation of the Gordon–Taylor equation for direct prediction of the glass transition temperature of mixtures; estimation of the crossover temperature of fragile glass formers; physical basis of the “Rule of 2/3”

Abstract

Predicting the glass transition temperature (T_g) of mixtures has applications that span across industries and scientific disciplines. By plotting experimentally determined T_g values as a function of the glass composition, one can usually apply the Gordon-Taylor (G-T) equation to determine the slope, k, which subsequently can be used in T_g predictions. Traditionally viewed as a phenomenological/empirical model, this work proposes a physical basis for the G-T equation. The proposed equations allow for the calculation of k directly and, hence, they determine/predict the T_g values of mixtures algebraically. Two derivations for k are provided, one for strong glass-formers and the other for fragile mixtures, with the modeled trehalose-water and naproxen-indomethacin systems serving as examples of each. Separately, a new equation is described for the first time that allows for the direct determination of the crossover temperature, T_x, for fragile glass-formers. Lastly, the so-called "Rule of 2/3", which is commonly used to estimate the T_g of a pure amorphous phase based solely on the fusion/melting temperature, T_f, of the corresponding crystalline phase, is shown to be underpinned by the heat capacity ratio of the two phases referenced to a common temperature, as evidenced by the calculations put forth for indomethacin and felodipine.