Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов

the liquid must undergo a phase transition before reaching it. This is, of course, the glass transition. In its original form, Kauzmann's paradox implicitly neglects possible differences in vibrational entropy between the amorphous and crystalline phases. This simplification is actually incorrect but it does not detract from the gist of the argument, for taking into account such differences would only shift T_K slightly. A more rigorous statement of the paradox is that the catastrophe would occur when the configurational entropy of the supercooled liquid vanishes.

      2.4.6 Potential Energy Landscape: Ideal Glass and Fragility

      Among the great many statistical mechanical models that have attempted to account for the glass transition and solve Kauzmann's paradox, the early one proposed by Gibbs and Di Marzio [48] is of special interest. It predicts that the supercooled liquid would transform to an ideal glass through a second‐order transition at the temperature T₀ at which its configurational entropy would vanish. Since then, the existence and the nature of such a transformation have been much debated. This debate notwithstanding, the important point for our discussion is the result subsequently derived by Adam and Gibbs [49] on the basis of a lattice model of polymers. This result is a very simple relationship between relaxation times and the configurational entropy of the melt, viz.

      (10)

      where A_e is a pre‐exponential term and B_e is approximately a constant proportional to the Gibbs free energy barriers hindering the cooperative rearrangements of the structure.

      Qualitatively, this theory assumes that structural rearrangements would be impossible in a liquid with zero configurational entropy so that relaxation time would be infinite. If two configurations only were available for an entire liquid volume, mass transfer would require a simultaneous displacement of all structural entities. The probability for such a cooperative event would be extremely small, but not zero, and the relaxation times would be extremely high, but no longer infinite. When configurational entropy increases, the cooperative rearrangements of the structure required for mass transfer can take place independently in smaller and smaller regions of the liquid.

Within this picture, relaxation is determined by the topology of potential energy wells in an n‐dimensional space and, particularly, by the density and relative depths of these wells as may be illustrated in a 1‐d representation of such a potential‐energy landscape (Figure 18, insets). A simple distinction can then be made between strong and fragile liquids [43, 50]. For the former, a low density of wells translates into a small configurational heat capacity and entropy and thus, in small departures from an Arrhenian temperature‐dependence of relaxation times as given by Eq. (7); for the former, the high density of wells is in contrast associated with high configurational heat capacities and entropies, and marked deviations from Arrhenian temperature dependences.

Owing to the simple proportionality between relaxation times and viscosity, this difference may be simply visualized in plots of viscosities as a function of T_g /T where T_g is the standard glass‐transition temperature [51]. A well‐known sketch (Figure 21) illustrates the point for a variety of inorganic and organic glass‐forming liquids [52]. As particularly exemplified in Chapter 4.1, this duality between fragile and strong liquids will be a recurrent theme in many other chapters of the Encyclopedia to which the reader is thus referred.

Figure 21 Fragility as a measure of the extent of temperature‐induced configurational changes in inorganic and organic glass‐forming liquids: correlations between relative C_p increases at the glass transition and deviations of viscosities from Arrhenius laws

      (Source: After [52]).

      Lower panel, from top to bottom: SiO₂, GeO₂, BeF₃, ZnCl₃, LiCH₃COO, 4 Ca[(NO₃)]₂·4 H₂O, o‐terphenyl, glycerol (C₃H₈O₃), and H₂SO₄·3 H₂O.

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