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

constraints are broken within structural units, the polyhedra become distorted. The existence of an extended TD network of distorted AB_V polyhedral units (where the BAB angular constraint is broken at the A site but the AB length constraint remains intact) is demonstrated by the example of glass formation in the CaO–Al₂O₃ binary system. Since neither component is a network former, glass formation in this binary system is poor. If the presence of CaO stabilizes four‐coordinated aluminum ions, Al^IV, with two AlO₄ tetrahedra sharing an oxygen vertex (as in silica), then the composition having 50% CaO should be a good glass former. However, experimental results show that glasses in this system form only in a small composition range at about 65% CaO [18]. It is possible to rationalize this observation with the constraint theory. Recently, Jahn and Madden [19] have reported from MD simulations that at 2350 K aluminum is present in Al₂O₃ melt in several different coordination states; about 54% Al^IV, 41% Al^V, 4% Al^VI, and 1% Al^III. Further, it was observed that some of the oxygens are present as O^III, oxygen coordinated by three aluminums (also known as oxygen triclusters), and the remaining as normal bridging oxygens, O^II. One can use this structural information and PCT to rationalize why the 65% CaO composition forms best glasses in this system. First, it can be assumed that the structures of Ca‐aluminate and alumina melts are similar except for the incorporation into the network of O from CaO. Next, one can simplify the structural information by neglecting the concentrations of Al^VI and Al^III. Let z be the fraction of Al^IV and (1 − z) that of Al^V. Then, for the composition x CaO·(1 − x)Al₂O₃, one shows with PCT that the degrees of freedom, f, is given by

(8)

When f = 0, Eq. (8) gives the following expression for the isostatic composition x* [x(f = 0)] in terms of z:

(9)

For z = 0.54, Eq. (9) gives x* = 0.65, the same value as for the best glass‐forming composition [18].

      3.4 Existence of Super‐Structural Units

      In B₂O₃ glass the presence of rigid, planar boroxol B₃O₆ units made up of three trigonal BO₃ units is well established [20]. In contrast to basic polyhedral units, AB_V, where a single A atom is coordinated by B atoms, super‐structural units such as boroxol units contain more than one A atom. Do super‐structural units then exist in other borate glasses? It is a question that has long persisted in the oxide‐glass science.

      Super‐structural units may be energetically more favorable in systems with long‐range interactions. However, their larger size raises difficulties to match the density of networks with the observed density of glass. For example, the boroxol units are topologically equivalent to the basic BO₃ trigonal unit (both have δ = 2 and V = 3). If all BO₃ trigonal units in B₂O₃ glass are replaced by B₃O₆ boroxol units, the length scale of the structural unit is doubled (the volume thus increasing by a factor of 8) while the mass of the unit is only tripled so that the density of a network of boroxol units is only 3/8 of that of a network of trigonal units. Further, based on topological considerations mentioned before, an extended 3‐D network can incorporate only a small fraction of super‐structural units with V > 4. For this reason, the di‐pentaborate and the di‐triborate groups, two of the six super‐structural units listed by Wright [20], probably do not exist in significant concentrations.

4 The Bond Constraint Theory

      As originally formulated by Phillips [1] for covalent networks, structural units are not considered in BCT. Instead, the system is viewed as a network of atoms at the vertices and covalent linear bonds at the edges. These covalent linear bonds provide r_i/2 linear constraints at the ith vertex of coordination number r_i. In addition, there also exist [r_i (d − 1) −d(d − 1)/2] covalent angular‐bond constraints at the ith vertex for a d‐dimensional network. The average number of constraints, n, per vertex is, therefore,

(10)

      where r is the average vertex coordination number. The condition of isostaticity (n = d) gives the following value for the critical coordination number r* (also called the rigidity percolation threshold):

(11)

Note that r* = 2 for d = 2 and r* = 2.4 for d = 3. It must be emphasized that Eqs. (10) and (11) assume that the angular constraints are intact at every vertex. This assumption does not always hold true as illustrated by silica where the angular constraints at oxygens are broken, which is generally the case for elements that do not belong to groups III, IV, and V and do not exhibit sp⁽ⁿ⁾ hybridization.

      Application of BCT to non‐covalent systems with long‐range interactions such as ionic systems is approximate at best, and questionable most of the time, because these systems do not lend themselves to the count of simple nearest‐neighbor constraint. For ionic systems, it is thus preferable to use PCT with structural units defined by the radius ratio of cations to anions.

      4.1 Self‐organization and the Intermediate Phase

Self‐organization designates chemical and/or topological rearrangements in a network that take place spontaneously to reduce the overall energy in the system [21]. An important consequence of this process is that it allows a system to exist as an isostatic network over a range of coordination numbers or compositions. This range is sometimes known as the intermediate phase or reversibility window [22]. The range depends on the system considered and, to some extent, on its thermal history as well as on the property being measured (e.g. enthalpy release during relaxation, Raman frequency shifts in glasses, or activation energy of viscosity). For example, a coordination number range from 2.39 to about 2.52 has been reported for the intermediate phase in Ge_xSe_(1−x) system from Raman frequency shifts [22], and a range from r = 2.35 to about 2.45 in the (Na₂O)_x(SiO₂)_(1−x) system from enthalpy relaxation [23]. Interestingly, Wang et al. [24] found no evidence of any intermediate phase in the Ge–As–Se system. Also, Shatnawi et al. [25] found no discontinuities or breaks but only smooth variation with respect to composition in the structural response of Ge_xSe_(1−x) glasses in the range 0.15 < x < 0.40 implying the absence of any phase transition associated with the start and end of the intermediate phase range.

      4.2 Non‐bridging Vertices (or Singly Coordinated Atoms)

      There has been much discussion in the literature [26] about the role of dangling vertices (or non‐bridging nodes) and their influence (if any) on the rigidity characteristics of a network. At least conceptually, it is clear that dangling vertices