Angell Plot
6.4 Estimate of Number of Correlated Molecules
Given that Ecoop(T) reflects the number of cooperatively moving molecules, which is expected to grow upon cooling, following a recent idea of N. Fatkullin
6.5. Conclusion 143
Figure 6.12: Correlation between the generalized fragility parameterµand the conventionally defined fragility index m; color code as in Figure 6.7; adapted with permission from ref. [8].
[46] one may estimate the numberNcorr by assuming thatEA =∈Ncoop(T) and further thatϵ∼kT holds. Thus, one gets
Ncorr=EA(T)/kT (6.3)
The result shown in Figure 6.13 shows a physically reasonable behavior, as at high temperatures,Ncoopis close to one and rise up to about 300 when the liquid is cooled down toTg. This result is similar to what has been reported very recently by the Augsburg group measuring the third-order non-linear dielectric susceptibility. [47] The latter allows to directly extractingNcoop. The authors have found that the apparent activation energyEA(T) scales withNcoop(T).
6.5 Conclusion
Combining different light scattering techniques (DM, TFPI and PCS), the evo-lution of the susceptibility spectra has been measured for a series of molecular glass formers and for temperatures between the boiling point (T ≤440 K) and Tg. The Tg values range from 92 K to 333 K. Comparing the obtained broad band LS spectra with those from dielectric spectroscopy significant differences are observed regarding the secondary processes such as excess wing and β-process occurring at high frequencies. In the case of the low-Tg liquids a broad high-temperature interval has been identified for which the extracted time con-stants of the α-processτ(T) are well described by an Arrhenius temperature dependence down to τ ∼= 10−12s. Here, structural and microscopic dynamics have essentially merged,i.e., a two-step correlation function, typical for glassy dynamics, is not observed any longer, yet a clear cut onset temperature cannot easily be identified. A trend to a crossover to Arrhenius high-temperature de-pendence well above the melting point is also found for systems with higherTg
144 6. From Boiling Point Down toTg
Figure 6.13: Estimate of the numberNcoop of cooperatively moving molecules as a function of temperature by applying Eq. 6.3.
and also for the non-fragile liquids, but the high-temperature activation energy E∞cannot be accessed reliably in these cases.
Having at hand correlation times ranging from 10−12≤τ /s≤100 which cover the entire regime of the liquid’s dynamics, i.e., from simple liquid to glassy dynamics, we have introduced a three-parameter description to interpo-late τ(T). We note that most functions interpolating τ(T) so far have been restricted to τ >10−10s due to missing high-temperature data. The present approach decomposes the activation energy E(T) in a temperature indepen-dent high-temperature contribution E∞ and a temperature dependent part Ecoop(T), the latter follows essentially an exponential temperature dependence.
Introducing a generalized Angell plot, namely Ecoop/E∞ vs. T /E∞ indicates the possibility that a common intersection point for the data of the group of liquids defines a crossover temperatureTA, for which TA / E∞=b is a univer-sal constant. Still a reliable decomposition of E∞ and Ecoop(T) is only possi-ble provided that sufficient high-temperature data are availapossi-ble and only for high-fragility systems. Thus, the parameter b as well as a (cf. eq. 6.2) refers to an average over the ensemble of liquids investigated here and may change with future experimental data. The different molecular glass formers distin-guish themselves by a (generalized) fragility parameterµ, which actually varies only weakly except for glycerol and propylene glycol. We emphasize that the experimentally observed E∞ must not be associated with some energy bar-rier in the liquid. Yet, the Arrhenius law revealed at high temperatures has to be taken into account by any liquid theory. Our attempt suggests that the high-temperature activation energy E∞ defines the energy scale of the glass transition phenomenon. Thus, simple liquid dynamics and glassy dynamics ap-pear to be connected.
References 145
Acknowledgements
Financial support by Deutsche Forschungsgemeinschaft (DFG) through project RO 907/11 and RO 907/15 is appreciated.
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