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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, T_g). 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 T_g(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.