ABSTRACT
In the previous chapters we have been analyzing how phenomena occurring in glassy materials can be reproduced by means of very simple models starting from a couple of basic ingredients, such as the separation of timescales between fast and slow processes and some kind of collective process for the relaxation of the slow modes. The previous description, however, is limited to the search for the fundamental mechanisms behind the slowing down of the relaxation and the fall out of equilibrium of the slow degrees of freedom inducing the glass transition. Those models are very helpful because, since they are simple, a lot of computation can be carried out and a rather straightforward connection between basic mechanisms and glass behavior can be obtained. They cannot, however, explain how these mechanisms arise in real systems. To get this information one should try to devise models that are direct representations of the intermolecular forces and chemical properties of the components of glass formers in nature. Unfortunately, moving to the level of a more faithful microscopic description implies a substantial loss in the power of theoretical predictability, unless further assumptions are introduced and numerical simulations are carried out to guide our intuition of physical phenomena. In this chapter we will, indeed, show and discuss a very broadly diffused
method to approach the study of the glassy behavior of models that are more realistic with respect to those met in previous chapters. These are systems whose space of states is complicated, both diversified and highly degenerate, and whose dynamics becomes slower and slower as temperature decreases, eventually leading to an arrest, right because of the complexity in the organization of the states. The price to pay to implement this “rugged landscape” description, as we will see in detail, will be to assume the existence of a fictitious space of the states, somehow related to the original one, and study the dynamics and the related glassy properties in this substitutive ensemble. We
of the
will, thus, deal with a symbolic dynamics, whose equivalence to the original one is the fundamental assumption of the whole approach. The characteristics of a glassy system arise from the complex topography
of the multidimensional function representing the collective potential energy yielding a nontrivial partition function and thermodynamic potential. The spatial atomic patterns in crystals and in amorphous systems share the common basic attribute that both represent minima in the free energy. At low enough temperature, where vibrations are minimal, one can try to assume that they are approximately represented by the minima of the potential energy function describing the interactions. The presence of distinct processes acting on two different timescales would mean that the deep and wide (i.e., wider than the crystal ones) local minima are geometrically organized to create a two length-scale potential energy pattern. Lowering the temperature of the liquid glass former, the bifurcation takes place as soon as it becomes “viscous.” By definition the temperature at which it occurs is the dynamic glass transition temperature Td (Sec. 1.1). The viscosity above which the decoupling of timescales occurs is usually estimated of the order of 10−2 Poise. This has to be compared with the order of magnitude, η ∼ 1013 Poise, at which the glass transition temperature, Tg, is operatively defined, allowing for a probe range for theories for the glass formation of about fifteen orders
In the general case, decreasing the temperature, the free energy local minima can, in principle, be split into smaller local minima or disappear. However, if we can assume that the possible birth/death of minima is not so dramatic that they lose their identity almost everywhere in the configurational space, we can set a one-to-one correspondence between metastable states and inherent structures (IS) [Stillinger & Weber, 1982, 1984; Stillinger, 1995; Sastry et al., 1998], i.e., between the minima of the free energy and the minima of the potential energy (see Fig. 6.1). Upon such an assumption one can, therefore, study the dynamical evolution of a glass former in its equilibrium and aging regime by means of a symbolic dynamics through ISs, rather than the true dynamics through metastable states at finite temperature. In this point of view an approximate approach to the problem is to divide
the complicated multidimensional landscape in structures formed by large deep basins and to describe the dynamics of the processes taking places as intra-basin and inter-basin. The potential energy landscape derived this way is, indeed, a description viewpoint. It helps to classify some static and kinetic phenomena associated with the glass transition according to a topographic analysis of the potential energy function. It was initially devised by Goldstein [1969] as an alternative approach to the study of the glassy state, that was potentially able to overcome the problems and inconsistencies of the free volume theory [Williams et al., 1955; Ferry, 1961],1 and the descriptive limitations of
of magnitude (see Chapter 1, in particular Secs. 1.1, 1.2).
