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6 Glass‐Liquid Transition
ОглавлениеStructural theories with energetic and microstructural criteria such as topological constraints describe elements that favor glass formation, i.e. the preservation of a topologically disordered distribution of basic elements in glasses. Kinetic theory shows how to avoid crystallization rather than explaining why the vitreous state really forms through the liquid–glass transition – it is at Tg that the “drama” occurs! Although kinetically controlled, the glass transition manifests itself as a second‐order phase transformation in the sense of Ehrenfest classification. Depending on the kind of measurement performed, it is thus revealed either as a continuous change of first‐order thermodynamic properties such as volume, enthalpy, entropy, or as a discontinuous variation of second‐order thermodynamic properties such as heat capacity or thermal expansion coefficient across the glass transition range.
As indicated by its name, the CPT treats the glass transition as a percolation‐type second‐order transformation [27]. It pictures it as the disappearance in the glassy state of percolating clusters of broken bonds – configurons. Above Tg, percolating clusters, which are formed by broken bonds, enable a floppier structure and hence a greater degree of freedom for atomic motion so that it results in a higher heat capacity and thermal expansion coefficient. Below Tg there are no extended clusters of broken bonds such that the material has acquired a 3‐D structure with a bonding system similar to that of crystals except for lattice disorder. This disordered lattice then contains only point defects in the form of configurons. Agglomerates of fractal structures made of these broken bonds are present only above Tg, which is given by:
(10)
In this equation Hd and Sd are the quasi‐equilibrium (isostructural) enthalpy and entropy of configurons present in Eq. (7) and ϕc is the percolation threshold, i.e. the critical fraction of space occupied by spheres of bond‐length diameters located within the bonding sites of the disordered lattice.
For strong melts such as SiO2, the percolation threshold in Eq. (10) is given by the theoretical (universal) Scher–Zallen critical density ϕc of 0.15 ± 0.01, which results in a practical coincidence between the calculated and measured Tg values. The parameter Hm has no influence on Tg as it characterizes the mobility of atoms or molecules through the high‐temperature fluidity of the melt – see Eq. (7). Because Hd is half of bond strength (Table 2), Eq. (10) shows that the higher this strength, the higher Tg. The vacancy model of the generalized lattice theory of associated solutions provides direct means to calculate thermodynamic properties as well as the relative number of bonds formed in glasses and melts when the second coordination sphere of atoms is taken into consideration [28].
In terms of chemical bonds, an amorphous material transforms to a glass on cooling when the topology of connections changes (Table 3), i.e. when the Hausdorff dimensionality of broken bonds changes from the 2.5 value of a fractal percolating cluster made of broken bonds to the zero value of a 3‐D solid. In terms of bonding lattice, the transition from the glass to the liquid upon heating may be explained as a reduction of the topological signature (i.e. Hausdorff dimensionality [29]) of the disordered bonding lattice from 3 for a glass (3‐D bonded material) to the fractal Df of 2.4–2.8 of the melt. These are the main changes that account for the drastic variations in material properties at glass‐to‐liquid transition [27].
Table 2 Classification of cations according to Diezel's field strength.
| Element | Valence Z | Ionic distance for oxides, Å | Coordination number | Field strength, 1/Å2 | Bond strength, kJ/mol | Function |
|---|---|---|---|---|---|---|
| Si | 4 | 1.60 | 4 | 1.57 | 443 | Network formers: F~1.5–2.0 |
| B | 3 | 1.50 | 3 | 1.63 | 498 | |
| 4 | 4 | 1.34 | 372 | |||
| P | 5 | 1.55 | 4 | 2.1 | 368–464 | |
| Ti | 4 | 1.96 | 4 | 1.25 | 455 | Intermediates: F~0.5–1.0 |
| 4 | 1.96 | 6 | 1.04 | 304 | ||
| Al | 3 | 1.77 | 4 | 0.96 | 335–423 | |
| 3 | 1.89 | 6 | 0.84 | 224–284 | ||
| Fe | 3 | 1.88 | 4 | 0.85 | ||
| 3 | 1.99 | 6 | 0.76 | |||
| Be | 2 | 1.53 | 4 | 0.86 | 263 | |
| Zr | 4 | 6 | 0.84 | 338 | ||
| 4 | 2.28 | 8 | 0.77 | 255 | ||
| Mg | 2 | 2.03 | 4 | 0.53 | ||
| 2 | 2.10 | 6 | 0.45 | 155 | Network modifiers: F~0.1–0.4 | |
| Pb | 2 | 6 | 0.34 | 310 | ||
| 2 | 2.74 | 8 | 0.27 | 151 | ||
| Ca | 2 | 2.48 | 8 | 0.33 | 134 | |
| Sr | 2 | 2.69 | 8 | 0.28 | 134 | |
| Li | 1 | 2.10 | 6 | 0.23 | 151 | |
| Na | 1 | 2.30 | 6 | 0.19 | 84 | |
| K | 1 | 2.77 | 8 | 0.13 | 54 | |
| Cs | 1 | 12 | 0.10 | 42 |
Table 3 Hausdorff dimensionality of the bonding system at glass transition.
| Amorphous material | Below Tg (glasses) | Above Tg (supercooled melts) |
|---|---|---|
| Broken bonds – configurons | 0 | 2.5a |
| Chemical bonds backbone cluster | 3 | 3 |
| Chemical bonds | 3 | 2.5a |
a Experimental dimensionality – 2.4–2.8.
Most experimental Tg data have been obtained by differential thermal analysis (DTA), differential scanning calorimetry (DSC), or dilatometry [30], where Tg is generally defined as the temperature at which the tangents to the glass and liquid curves of the relevant property intersect (Chapter 3.2). Heating (cooling) rates for DTA/DSC measurements are typically as high as 10 K/min whereas they are in 3–5 K/min range in dilatometry. As already stated, the glass transition is not abrupt but typically occurs over a few tens of degrees. For not very high cooling rates (q), its dependence on q is given by the Bartenev–Ritland equation:
(11)
where a1 and a2 are empirical constants. Although Eq. (11) also results from CPT, it should be replaced by a generalized version at high cooling rates [31]. In addition, CPT predicts that the transition takes place not as a sharp discontinuity, but over a finite temperature interval where the properties of the material depend on time as well as on thermal history.
Following the analysis of [8], we may ask why viscous liquids eventually vitrify instead of remaining in the supercooled liquid state when they escape crystallization. One answer to this question is purely kinetic and relies only on increasingly long relaxation times or increasing viscosities on cooling. The glass transition would result only from the limited timescale of feasible measurements so that any glass would eventually relax to the equilibrium state if experiments could last forever. In fact, a simple thermodynamic argument proposed by Kauzmann [32] indicates that this answer is incorrect. The reason originates in the existence of a configurational contribution that causes the heat capacity of a liquid to be generally higher than that of a crystal of the same composition. As a consequence, the entropy of the liquid decreases on cooling faster than that of the crystal (Figure 4).
If the entropy of the supercooled liquid were extrapolated to temperatures much below Tg, it would become lower at a temperature TK than that of the crystal. Because it is unlikely that an amorphous phase could ever have a lower entropy than an isochemical crystal, the conclusion known as Kauzmann's paradox is that an amorphous phase cannot exist below TK. The temperature of such an entropy catastrophe constitutes the lower bound to the metastability limit of the supercooled liquid. As internal equilibrium cannot be reached below TK , the liquid must undergo a phase transition before reaching this temperature. This is, of course, the glass transition, and Kauzmann's paradox suggests that, although it is kinetic in nature, it anticipates a thermodynamic transition. In other words, CPT treats the glass transition as a true phase transformation although as a nonequilibrium one. The liquid transforms in a continuous way into a glass, which behaves mechanically as a crystalline solid when the motions of atoms become very much frustrated below Tg where the extensive clusters of broken bonds of the liquid are no longer present. The degree of frustration then is actually the same as in a 3‐D crystalline material so that the heat capacity does not show the same high rate of change as in the liquid. This feature is clearly seen both in experiments and as an outcome of the CPT concept (Figure 5). Importantly, CPT yields a universal law for susceptibilities such as heat capacity or thermal expansion near Tg [3, 27]:
Figure 4 Entropy of the amorphous and crystalline phases of diopside, CaMgSi2O6.
Source: After [8].
The liquid transforms into a glass below Tg, therefore the entropy of condensed phase (upper curve) does not follow the dashed line which is an extension of liquid entropy curve below Tg.
Figure 5 Comparison between the heat capacities of amorphous o‐terphenol measured and calculated with configuron percolation theory.
Source: After [3].
(12)
A last feature deserving to be mentioned is the “universal” dependence of the light scattering intensity on the time after a temperature jump in the glass transition range of oxide glasses, which is known as the Bokov effect [33]. The intensity displays a maximum whose height and location on the timescale depends on the previous history of the glass. The Bokov effect is associated with nonequilibrium fluctuations produced by coupling between hydrodynamic modes. Detailed investigations in the past decade have demonstrated that similarities observed in the glass transition region of oxides and polymers account for structural transformations related to the formation of spatially extensive structures, which in turn could be related to clustering effects similar to that envisaged by CPT and other similar models. The Bokov effect thus is providing additional arguments to characterize the glass transition as a second order like phase transformation rather than simply as a slowing down of dynamic processes.
