Excluded volume effect on entropy and clustering behaviors in supercooled liquids and glasses

We study the significant entropy changes in supercooling and glass transition by developing a two (H and L) state cluster model, where the H-state represents the clusters with the coordination number z being larger than the reference size zR and the L-state those with z<zR, respectively. We exploit the excluded volume effect on the clustering behaviors of various glass-formers and the temperature dependence (at fixed pressure of 1 atmosphere) of the entropy difference Δso between the two cluster states on the basis of zR=1+zRo[exp(−Δso/kB)−1], where zRo=1.8 as taken universally for all the glass-formers investigated. The temperature dependence of the average size 〈zc∗〉 of clusters with z≥zR is then determined for some representative glass-formers, such as GeO2, 1-propanol, glycerol, propylene carbonate, o-terphenyl, and so forth, covering the entire fragility spectrum studied experimentally. As a result, the temperature dependence of 〈zc∗〉 is found to highly depend on the fragility index ϑ associated with the temperature dependence of the excluded volume B∗(ϕ,T) of which statistical mechanical representation is given by the generic van der Waals equation of state. With decreasing T toward the glass transition temperature Tg, 〈zc∗〉 increases to attain universally about 13 at Tg regardless of the fragility and molecular details of the glass-formers. We specifically analyze in depth the theoretical predictions for the cluster sizes and the experimental estimations for the length scales associated with temporary clusters of slow segments in supercooled glycerol as about 1.4±0.5 nm for glycerol at 10 K above Tg. Given the large uncertainties associated with the experiment, the present theory appears to be in good agreement with the experiment.