Constraint Theory of Inorganic Glasses
Prabhat K Gupta
The Ohio State University, Columbus, OH, USA
1 Introduction
Compositional innovation has been the bedrock of glass R&D for more than a century. Two questions always come up when searching for a new composition: Will the material be easy to form into glass from the liquid and will its properties have the values desired for the application considered? This set of properties includes not only those of the finished glass product (mechanical, for example) but also those necessary for successful processing (such as melt viscosity and glass transition temperature, Tg). In principle, the topological constraint theory (TCT) can aid in providing answers to both of these questions.
Historically, TCT grew along two distinct paths, both starting at about the same time in the late 1970s. The more widely known view, termed the bond constraint theory (BCT), was formulated in 1979 by Phillips [1] and is most useful for chemically disordered covalent systems such as chalcogenides. The other view applies to chemically ordered systems such as oxide glasses whose structure, à la Zachariasen ([2], Chapters 2.1 and 3.1), consists of topologically disordered extended networks of corner‐sharing rigid polyhedra. This view was developed by Cooper [3] in 1978 for two‐dimensional networks and later extended to three‐dimensional networks by Gupta and Cooper [4]. We refer to this view as the polyhedral constraint theory (PCT).
During the early development, constraints were counted as either intact (=1) or broken (= 0) and temperature (T) played no role. In 1993 [5] (and in more detail in 1999 [6]), Gupta introduced the concept of temperature‐dependence of bond constraints. In 2009, Gupta and Mauro applied the T‐dependent BCT to rationalize the composition (x) dependence of the glass transition temperature Tg(x) in binary chalcogenide [7] and in binary oxide systems [8]. Later, Bauchy and Micoulaut [9] validated the phenomenology of T‐dependent bond constraints with molecular dynamics (MD) simulations.
With growing interest in TCT, much effort has been and is being invested in applying it to model the composition dependence of a variety of properties in glassy systems. The most successful thus far has been the work of Mauro and colleagues [10] for the composition dependence of the room‐temperature hardness of oxide glass systems. Not being a comprehensive review, however, the present chapter does not include these recent applications. It is organized as follows: an introduction to TCT is presented in the Section 2. It establishes key definitions and terminology and outlines the underlying conceptual framework. In Section 3, elements of PCT and some of its applications to oxide glass‐forming systems are presented. Section 4 does the same for BCT with examples from chalcogenide systems. In Section 5, we discuss the phenomenology of the temperature dependence of constraints – a development that has generated much excitement owing to its remarkable ability to model variations of properties with composition. Some fundamental issues associated with TCT are discussed in Section 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
