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system [10].

Figure 4 Composition dependence of the glass transition temperature for the (a) sodium borate and (b) lithium borate systems. Solid curves: predicted T_g(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‐T_g Regimes

      A generalized T‐dependent activation energy, H(T), is defined as the slope of the Arrhenius plot of viscosity:

      (23)

The ratio, H(T_g, x)/(k_B T_g), 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 T_g appears to be a constant function of composition for a small composition range. They called this composition range the “iso‐T_g regime” (Figure 4). The iso‐T_g step results when a bond constraint breaks exactly at T_g. It can be shown that, within the iso‐T_g 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‐T_g 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, x_f is the fraction of floppy modes (= f/3); p_j and q_j are, respectively, the momentum and position coordinates of oscillators representing vibrational modes of frequency ω_j and floppy modes of frequency ω_o. 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 ω_o (<<ω_vib) 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 ω_D 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, ΔC_V, is given by

      (27)

      where x = h ω_o/kT.

      As Naumis has pointed out, the physical picture in the PEL is clear. Since ω_o 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