A brief discussion: Thermodynamic and dynamic fragilities, non-divergent dynamics and the Prigogine–Defay ratio Gregory B. McKenna * Department of Chemical Engineering, Texas Tech University, Lubbock, TX 79409-3121, USA Laboratoire de Physicochimie des Polymères et Milieux Dispersés, UMR 7615/UPMC/CNRS/ESPCI, 10 rue Vauquelin, 75231 Paris cedex 05, France article info Article history: Available online 9 April 2009 PACS: 61.20.Lc 64.70.P- 64.70.pe 64.70.ph 64.70.pj 66.20.Ej 83.80.Ab Keywords: Glass transition Polymers and organics Viscosity and relaxation Fragility Rheology Stress relaxation Structural relaxation Viscoelasticity abstract This article brings together some of the work performed by the present author and collaborators that is related to the glass transition event and to some of its entropy aspects. The purpose of the work and dis- cussion was motivated by a view that some of the frameworks in which we currently look at glassy behavior, while potentially useful, may also have limitations that we often do not fully consider. Discus- sion focuses on isochoric glass formation paths, thermodynamic and dynamic fragilities and how dynamic fragility in many systems (especially polymers, metals, ionic liquids and hydrogen bonding sys- tems) seems to vary primarily with the glass transition temperature itself. This leads to the conclusion that such systems have an apparent activation energy that varies as the square of the glass temperature. The work then discusses evidence for a non-diverging relaxation time or viscosity as the glass tempera- ture is approached and ends with a discussion of the Prigogine–Defay ratio. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction This article brings together some of the work performed by the present author and collaborators that is related to the glass transi- tion event and to some of its entropy aspects. The purpose of the work and discussion was motivated by a view that some of the frameworks in which we currently look at glassy behavior, while potentially useful, may also have limitations that we often do not fully consider. The work begins with a semi-chronological discus- sion of the author’s own endeavors that are related to fragility in glass-forming liquids with the recognition that such work has motivated other works and, particularly the aspects related to the correlation between dynamic fragility and the heat capacity jump at the glass temperature may have been partially resolved by others. Because of the chronology, however, the work presented focuses on the original thoughts and mentions more recent works that partially go beyond these. Subsequent work by the author sug- gests that a large portion of the dynamic fragility is a simple link- age with the glass transition temperature itself and this is discussed and some recent works that take a different approach are mentioned. Discussion of the behavior of the dynamics of glass-forming systems would be incomplete without a consideration of the wide observation of the super-Arrhenius temperature dependence of the dynamics and observations made by this author and others that suggest that many systems may actually show discrepancies from this picture with a turnover of the behavior back towards an Arrhe- nius-like temperature dependence when the equilibrium liquid is probed below the nominal (100 s or calorimetric) glass tempera- ture. Clearly, non-diverging time scales do not fit with our current glass paradigm, but experimental observations suggest that actual divergence is not achieved. The author’s own work is discussed in context with the works of others. Also, the question of the Prigogine–Defay ratio R is discussed. It is this author’s view that the value of R is defined for a thermody- namic system in equilibrium. If this is the case, the Prigogine–De- fay ratio is a simple tautology and must be equal to unity. In fact, even in non-equilibrium situations the value must equal unity 0022-3093/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2008.11.023 * Address: Department of Chemical Engineering, Texas Tech University, Lubbock, TX 79409-3121, USA. E-mail address: greg.mckenna@ttu.edu Journal of Non-Crystalline Solids 355 (2009) 663–671 Contents lists available at ScienceDirect Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/locate/jnoncrysol