Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов
6 along with suggestions for possible ways to embed the TCT phenomenology within the general framework of the potential energy landscape (PEL) of liquids and glasses.
Not many comprehensive reviews of TCT are available in the literature. Early on, Phillips [11] published an introductory paper on BCT entitled “The Physics of Glass.” Thorpe [12] provided an account of the early developments of the rigidity percolation theory. Gupta [5] reviewed the PCT formalism in 1993 and later more extensively in 1999 [6]. Recently, Naumis and Romero‐Arias [13] have reviewed the physics of the constraint theory, its connection with thermodynamics and with glass transition. More recently, Mauro [14] has published an excellent introduction on TCT.
List of Acronyms
TCTtopological constraint theoryBCTbond constraint theoryPCTpolyhedral constraint theoryPELpotential energy landscapeTDtopologically disordered
2 Concepts of the Topological Constraint Theory
Network‐based concepts enter naturally in TCT, which treats a glass‐forming system as an atomic network where atoms constitute the nodes (or vertices) and atomic interactions (i.e. chemical bonds) constitute the edges (or linkages) of the network. The properties of an atomic network depend both on its chemistry (how atoms of different species are placed on its nodes) and on its topology (how various nodes are interconnected without regard for the nature of atoms).
2.1 Network Chemistry
2.1.1 Composition of an Atomic Network
When a network contains only one kind of atom (i.e. the network is chemically homogeneous), it is called one component (or unary). Networks are termed binary when they contain only two kinds of atoms, and multicomponent if three or more distinct atoms are present. The (molar) fractions {xi} of atoms of the different components specify the average chemical composition of a network. In this paper, we discuss only unary or binary networks of type AxB(1 − x), where x represents the mole fraction of A atoms.
2.1.2 Chemical Order and Disorder
For a given x, the arrangement of A and B atoms on the vertices of a network determines the nature of the chemical order present. Often, for networks having compound‐like compositions, ACBV (where C and V are integers), the arrangement of two types of atoms is well defined: every A atom is coordinated by a number, V, of B atoms. Similarly each B atom is coordinated by another number, C, of A atoms. Such networks with well‐defined chemical arrangement are called chemically ordered. In these networks, linear bonds exist only between dissimilar atoms (i.e. heteropolar bonds). A prime example of such networks is that of silica (SiO2): each silicon is coordinated by four oxygens (V = 4) and each O is linked to two silicons (C = 2).
Not all glasses can be chemically ordered as this state can be achieved only for some definite stoichiometries. Noncompound‐like compositions are thus necessarily disordered. A signature of chemical disorder is the presence of homopolar bonds (between similar atoms). Even compound‐like compositions can be chemically disordered in covalent systems because of the presence of a significant fraction of such homopolar bonds: an example is GeSe2 [15].
2.1.3 Atomic Interactions and Chemical Bonds
Bonds constituting the edges of an atomic network represent simple mappings of atomic interaction potentials among the constituent atoms. For the purpose of TCT:
1 Pair interactions are mapped into narrow potential wells, forming linear bonds of fixed length between neighbors. The tails of pair interaction potentials are ignored. Whereas this assumption is reasonable for short‐range interactions, its validity is questionable for long‐range coulombic potential, especially in BCT.
2 Among many‐body interactions, only triplet interactions are retained. They, together with next‐nearest neighbor pair interactions, form the angular bonds at the vertices. Whereas angular constraints are important in BCT, they do not play any direct role in the PCT.
2.1.4 Structural Units in Chemically Ordered Networks
In a binary chemically ordered network, each A atom is coordinated by exactly the same number (V) of B atoms. Hence, the network can be viewed as made up of well‐defined polyhedral (ABV) structural units having an A atom in the center and a B atom at each vertex (i.e. corner). Further, C such structural units are connected in the network to each B atom. The chemical formula of the structural unit, AB(V/C), is therefore the same as that of the network as a whole.
2.2 Network Topology
For topological considerations, an atomic network is treated as if the observer is “chemically blind.” In other words, the network is considered simply as a combination of vertices and edges disregarding the nature of the atoms occupying the vertices. Thus, topology refers only to how the nodes are interconnected in a network.
Local (or short‐range) topology at the ith vertex is described by the number (ri) of edges shared by that vertex (i.e. the vertex coordination number). A vertex having ri = 1 is called non‐bridging or dangling. Vertices with ri > 1 are called bridging (or network‐forming). When ri is the same for all vertices, the network is called regular. Otherwise, it is irregular. For an irregular network, the average vertex coordination number, r, also called the connectivity, provides a measure of its short‐range topology. Note that, in chemically ordered networks, r is related to the two coordination numbers V and C by
(1)
The intermediate‐range topology of a network is characterized by its ring‐size distribution so that it, for instance, depends on whether neighboring structural units share edges or corners. By definition, noncrystalline networks have no long‐range topological order. For this reason, they are termed topologically disordered (TD) networks [4].
2.3 Bond Constraints
The edges of a network represented by linear bonds constitute linear constraints on the coordinates of the vertices. In atomic networks, angular bonds give rise to additional constraints at the vertices. The linear and angular constraints, together, are called bond constraints. Since an angular constraint can be viewed simply as the result of an additional cross‐linear bond between next‐nearest neighbors, the linear and angular constraints carry equal weights. In other words, during constraint counting, one angular constraint and one linear constraint add up to two constraints.
It is important to distinguish between independent and dependent (or redundant) constraints. Constraints in a network that do not change its deformation behavior are called dependent. Consider, for example, a finite planar network of four nodes situated at the corners of a square (Figure 1) for which the sides constitute four linear constraints. If these are the only constraints present, the network is floppy (i.e. it can be deformed). When a diagonal constraint is added, however, the network becomes rigid. When a second diagonal is added as the sixth constraint, no further change occurs in the deformation behavior of the network – it remains rigid. The sixth constraint, in this example, is therefore a dependent constraint. In TCT, it is important to count only the independent constraints and to exclude the dependent. To determine whether a constraint is dependent or not can be a challenging task. Owing to the presence of long‐range topological order, crystalline networks contain a significant number of dependent constraints. Whereas one has to be extremely careful