Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов
system [10].
Figure 4 Composition dependence of the glass transition temperature for the (a) sodium borate and (b) lithium borate systems. Solid curves: predicted Tg(x) using the temperature‐dependent TCT. Points: experimental data
(Source: From [8]).
Figure 5 Variation of fragility with composition in the (a) sodium borate and (b) lithium borate systems. Curves calculated with the T‐dependent TCT. The step increase in fragility around x = 0.2 is a consequence of a fragility transition in these systems
(Source: From [8]).
5.5 Fragility (or Rigidity) Transitions and Iso‐Tg Regimes
A generalized T‐dependent activation energy, H(T), is defined as the slope of the Arrhenius plot of viscosity:
(23)
The ratio, H(Tg, x)/(kB Tg), is proportional to the fragility m. In some systems, the activation energy (or fragility) shows rounded discontinuities as a function of T or as a function of composition, X. These jumps are referred to as fragility transitions. An example of such transition in the alkali‐borate systems is shown in Figure 5. Fragility transition as a function of temperature for a fixed composition is illustrated in Figure 6. In a temperature‐induced fragility transition, a system always becomes more fragile at higher temperatures simply because more constraints are broken as the temperature is increased.
Mauro et al. [8] made an interesting observation in the binary alkali‐borate melt xM2O·(1 − x)B2O3 systems where Tg appears to be a constant function of composition for a small composition range. They called this composition range the “iso‐Tg regime” (Figure 4). The iso‐Tg step results when a bond constraint breaks exactly at Tg. It can be shown that, within the iso‐Tg regime, the fragility remains constant and equal to the low value of about 16 that is observed for strong glasses. Interestingly, the composition range of the iso‐Tg regime (at least in the alkali‐borate system) is nearly the same as that of the reversibility window observed by Boolchand and colleagues [22]. This coincidence raises the possibility of some connection between the two phenomena, an area that requires further investigation.
Figure 6 Contrast between strong (SiO2) and highly fragile (O‐Terphenyl, OTP) in plots of viscosity against reciprocal temperature, and intermediate cases of liquids fragile at high temperatures and becoming strong at low temperatures (water, glass‐forming metals) through a diffuse transition (cf. Chapter 3.8;
Source: From [35]).
6 Topological Constraint Theory, Thermodynamics, and the Potential Energy Landscape Formalism
The impressive applications of TCT demand answers to fundamental questions such as how is TCT connected to the thermodynamics of liquids and glasses and how to formulate TCT from the first‐principles statistical physics of potential‐energy landscapes of liquids and glasses. This is an area that has not received much attention so far except for the work of Naumis and coworkers [13, 32] that we summarize in this section.
Naumis uses simple harmonic potentials to express the Hamiltonian (H) of a floppy system as follows:
(24)
Here, xf is the fraction of floppy modes (= f/3); pj and qj are, respectively, the momentum and position coordinates of oscillators representing vibrational modes of frequency ωj and floppy modes of frequency ωo. For real systems, one has to use more sophisticated interaction potentials. Nonetheless, a harmonic model gives a reasonable qualitative feel of the thermodynamics of the floppy modes. Naumis assigns a small but finite frequency ωo (<<ωvib) to each floppy mode. The equilibrium thermodynamics of this Hamiltonian can be calculated easily. The result for the internal energy U of the system is
(25)
Here, D is the well‐known Debye function and ωD is the Debye frequency. According to Naumis, the expression for entropy is as follows:
(26)
The last terms in Eqs. (25) and (26) represent the contributions of the floppy modes and constitute the configurational energy and entropy, respectively. As discussed in Section 5.2, the configurational entropy is proportional to f.
From Eq. (25), the configurational heat capacity, ΔCV, is given by
(27)
where x = h ωo/kT.
As Naumis has pointed out, the physical picture in the PEL is clear. Since ωo is small, the curvatures of the potential energy surface at the inherent structures are small in the floppy directions. Naumis refers to these directions as channels in the landscape. The landscape is flatter in these channels, thus allowing greater configurational entropy and greater structural freedom upon increase in T and thus greater fragility.
7 Perspectives
Over