ABSTRACT
Below the temperature Tg, a vitreous liquid becomes solid. Even though its properties seem to be constant in time for any practical use, we have seen that the amorphous structure is actually not in an equilibrium state and we cannot refer to it as a thermodynamic state. Indeed, the glass and the liquid phase cannot be connected by any path in a time independent parameter space: no time independent thermodynamic transformation can ever bring a glass former from the liquid phase to the glass phase below Tg. Because of this ever-standing lack of equilibrium, the time will always play a fundamental role in the formation, in the description and in the properties of the glass. Nonequilibrium thermodynamic theories were worked out in the first half of
the last century. They apply to systems close to equilibrium. Typical applications are systems with heat flows, electric currents, and chemical reactions. Key players have been De Donder, who introduced the concept of “rate of reaction” for chemical reactions, his follower Prigogine, who wrote with Defay the book Chemical Thermodynamics [Prigogine & Defay, 1954] and received the Nobel prize in chemistry for his contributions to nonequilibrium thermodynamics, and Prigogine’s student Mazur, who spent most of his scientific career in Leiden (The Netherlands) and wrote with de Groot (Leiden, Amsterdam) the classical textbook Nonequilibrium Thermodynamics [de Groot & Mazur,
of the
1962]. The basic assumption in this field is the presence of local thermodynamic equilibrium, and the basic task consists in calculating the entropy production.3 For a modern review, we may mention Schmitz [1988]. The nonequilibrium thermodynamics we aim at should apply, instead, to
systems far from equilibrium. For example, neither a large nor a small piece of glass is in local thermodynamic equilibrium; in either cases a very long waiting time would be needed to reach equilibrium, much longer than the time of the experiment. Thermodynamics far from equilibrium is yet a mined field, still waiting for a comprehensive theory that could represent in a thermodynamic frame the most general aging systems, whose behavior is dominated by nonthermalized processes. Until very recently, some confusion has been caused by statements implying that thermodynamics had no applicability for glasses because these are intrinsically out of equilibrium. Actually, this point of view, that superficially neglects the possibility of nonequilibrium thermodynamics both near and far from equilibrium, is, in our opinion, arguable. Thermodynamics started out as a theory on the energy household of steam
engines. The founding paper by Nicolas Le´onard Sadi Carnot “Sur la puissance ” still stands as a benchmark paper in the field.4 The theory was born in the first half of the 19th century as a new way of looking at phenomena that, in contrast with the Newtonian mechanics approach, was not deterministic nor predictive, and whose goal was to establish the constraints imposed by nature in the exploitation of its forces, and to control and drive energy transformations in order to estimate the optimum performance of a thermal machine. The fact that the theory was later mainly developed at equilibrium does not mean that the equilibrium hypothesis is the fundamental issue of thermodynamics. The difficulties met so far in the attempt of using thermodynamics for glasses could be simply related to the unfounded equilibrium hypothesis. Many kinds of dynamics can occur in nature. Our aim is to find some
universality in glassy systems, to find at least a subset of variables for which their dynamics can be encoded in a thermodynamic framework. We should not want much more than this. The considered systems are, at the basic level, very intricate and too finely detailed mechanisms cannot expose thermodynamic behavior. It may very well happen that certain variables display some kind of thermodynamic behavior, while other variables of the same system do not.5
This is still a gain, as, in principle, no universality is expected.6
