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 the last 25 years, TCT has evolved from theoretical inquiries about the existence of Zachariasen's TD networks and easy glass formation in simple systems to a widely used scheme for modeling variations of properties with composition in multicomponent glasses and glass‐forming liquids. The trends predicted by TCT are in good agreement with experimental results. This is largely because TCT takes as input the observed or modeled variation of topology with composition and a fundamentally sound temperature‐dependence of chemical bond strengths. In this sense, TCT represents a major leap in the development of science‐based engineering of new glass compositions.
In spite of its success, several limitations and fundamental issues remain unresolved. We list some of these here.
1 For systems with short‐range interactions where BCT is applicable, it is unclear when to treat the angular constraints as broken (even at low temperatures). For example, why is the angular constraint broken at oxygens in silica? Are they broken at Se in selenides? It appears that angular constraints are intact primarily in group III, IV, and V elements that exhibit sp(n) hybridization. Clearly, a more detailed understanding of the basis of the angular constraints will lead to more accurate predictions of BCT.
2 Extension of TCT to systems with long‐range potentials (especially ionic interactions) remains vague and questionable at present. Whereas BCT applies only to systems with short‐range covalent interactions, PCT works better for ionic systems. The effects of chemical order in BCT and of chemical disorder in PCT need further examination. There is a clear need for a more fundamental basis for applying TCT to chemically disordered systems having long‐range interactions.
3 It has been noted that the rigidity percolation threshold may not occur at a single composition, but over a range of compositions because of either localization of constraints or localization of degrees of freedom in an otherwise isostatic matrix. Whereas a discussion of iso‐Tg range is presented, there does not exist at present a theory to determine the rigidity percolation composition range.
4 The role of intermediate‐range topology (ring‐size distribution) in determining the deformation of a network remains unclear. For example, edge‐sharing between tetrahedra in the Ge–Se system appears to have no effect on the BCT results since both edge and corner sharing give rise to identical short‐range topology (i.e. the same value of r).
5 Whereas an approximate relation between configurational entropy and degrees of freedom has been discussed, a more rigorous understanding of constraints (or degrees of freedom) in general in terms of features
