Diverging views on glass transition

Analysis of the best available data on the behaviour of a large number of glass-forming organic liquids suggests that the widespread belief that a glass ceases to flow below its transition temperature could be wrong.

The glass transition is one of the most intriguing phenomena in the world of soft condensed matter. Despite decades of study, many aspects of the behaviour of glass-forming liquids remain elusive. That the viscosity (or relaxation time) of a glass-forming liquid diverges to infinity at some finite temperature above absolute zero, is one thing, at least, on which most experts agree. But on page 737 of this issue, Hecksher and co-workers¹ present a serious challenge to even this aspect of common knowledge. By a systematic analysis of the temperature-dependent relaxation-time data from a range of organic glass-forming liquids, they suggest that the conventional equations used to describe their behaviour are inadequate, and that new theoretical frameworks are needed.

The dynamic divergence, or 'super-Arrhenius' behaviour, of glass-forming systems is commonly represented either by the Vogel–Fulcher–Tammann (VFT)^2,3,4 or Williams–Landel–Ferry (WLF)⁵ expression. The VFT and WLF equations are mathematically equivalent, the former being used principally by the molecular liquids community whereas the latter is frequently used by the polymer community. The VFT equation expresses the relaxation time (τ) or the viscosity (η) in a form that is similar to the Arrhenius equation, but with the important exception that it diverges exponentially at a finite temperature (T_∞). The paradigm that these equations represent has been around for over half a century, and the conclusions drawn by Hecksher et al. that they are inconsistent with the behaviour of real systems represents a significant shift in how we think of the glass-transition phenomenon. This prompts several questions. Must the diverging timescales contained within the VFT equation represent something essential to the nature of the glass transition? Must a theory of the glass transition, to be successful, accurately predict such behaviour? Or is all the essential physics contained with the VFT's description of the behaviour of glass formers at temperatures away from the point at which it seems to diverge?

Before addressing such questions directly, it is important to mention a few caveats. The temperature T_∞ has often been associated with the so-called Kauzmann⁶ temperature T_k, which itself has been linked to an ideal glass temperature, T₂. Accurate determination of the value of T₂ is extremely difficult because at temperatures below the laboratory T_g the time for a system to reach equilibrium becomes extremely long, and to the point that measurements simply become unfeasible. Consequently, the data Hecksher et al. analyse is collected from systems at temperatures above the glass-transition temperature, and so their conclusions cannot be considered as unequivocal. Moreover, although Hecksher et al. recognize the lack of exact coincidence between the values of T₂ and T_∞, the expected relationship between them remains a conceptually important one, and makes the paper's conclusions even more far reaching if confirmed.

Hecksher et al. suggest that glass formation in polymers may be fundamentally different from the liquid–glass transition. Had they not taken this stand, the state of the literature provides stronger support for their perspective. For instance, Fig. 1 shows a compilation of data for the segmental dynamics (those related to the glassy state in polymers) for polystyrene that includes literature date and new data from Simon et al.⁷, presented on a 'VFT-plot' of the logarithm of the normalized relaxation time (shift factor a_T = τ/τ_ref) against 1/(T–T_∞). If the dynamics were to be consistent with the VFT equation, the plot should be a straight line. If the deviation from linearity were upwards as T–T_∞ approaches zero, the divergence of timescales would be stronger than VFT, and if the divergence were downwards, it would suggest that the divergence is weaker than VFT or could be going towards Arrhenius-like. It is clear from the figure that the divergence is downwards and the data are fully consistent with conclusions made by Hecksher et al. — that is, these data too suggest that the timescales related to the glass transition may not diverge at finite temperatures. Similar findings, though presented differently, have been made for polymers and small-molecule glass-formers^8,9. However, this too should be taken with a grain of salt as not all investigations¹⁰ agree with these results, and there is evidence to suggest that different processes⁷ have different time- and temperature-dependencies near the glass transition — which is yet another confounding and important problem of glass-forming liquids.

Figure 1: VFT plot of segmental relaxation data (as shift factors a_T) for polystyrene.

Experimental and literature compilation from Simon and colleagues⁷. Reprinted with permission from ref. 7. © 2001 Elsevier.

To add to this, even the expected connection mentioned above between the Kauzmann temperature T_k and the VFT temperature T_∞ is far from clear. There is important work in the literature that challenges not only this link, but also whether or not T_k itself represents a meaningful or significant thermodynamic signature of the glass transition^11,12. It is likely, then, that the nature of the glass transition will remain a difficult problem for the foreseeable future. Despite the extremely long timescales necessary to conduct experiments at temperatures below the glass transition, it seems that such could be the only way to fully assess the validity of the conclusions reached by Hecksher et al. It may be that other methods of extrapolation to the equilibrium state need to be considered¹² as well. But at the very least, these conclusions do suggest that that the status quo represented by the VFT and WLF equations will need to be reconsidered if we are ever to reach a definitive theory for the glass transition in complex fluids.