Rigidity theory (physics)

Rigidity theory, or topological constraints theory, is a tool for predicting properties of glasses based on their composition. It was introduced by Phillips in 1979^[1] and refined by Thorpe in 1983.^[2] Inspired by the study of the stability of mechanical trusses as pioneered by James Clerk Maxwell, this theory reduces complex molecular networks to nodes (the atoms, proteins ) constrained by rods (chemical constraints), thus filtering out microscopic details that ultimately don't affect macroscopic properties. This has further implications such as in understanding adaptability in protein protein interaction networks.^[3]

In molecular networks, atoms can be constrained by radial 2-body bond-stretching constraints, which keep interatomic distances fixed, and angular 3-body bond-bending constraints, which keep angles fixed around their average values. As stated by Maxwell's criterion, a mechanical truss is isostatic when the number of constraints equals the number of degrees of freedom of the nodes. In this case, the truss is optimally constrained, being rigid but free of stress. This criterion has been applied by Phillips to molecular networks, which are called respectively flexible, stressed-rigid or isostatic when the number of constraints per atoms is respectively lower, higher or equal to 3, the number of degrees of freedom per atom in a three-dimensional system.^[4] Typically, the conditions for glass formation will be optimal if the network is isostatic, which is for example the case for pure silica.^[5] Flexible systems show internal degrees of freedom, called floppy modes,^[2] whereas stressed-rigid ones are complexity locked by the high number of constraints and tend to crystallize instead of forming glass during a quick quenching.

Hence, rigidity theory allows the prediction of optimal isostatic compositions, as well as the composition dependence of glass properties, by a simple enumeration of constraints. Notably, the theory played a major role in the development of Gorilla Glass 3.^[6] Extended to glasses at finite temperature^[7] and finite pressure,^[8] rigidity theory has been used to predict glass transition temperature, viscosity and mechanical properties.^[4] It was also applied to granular materials^[9] and proteins.^[10]

In 2001, Boolchand and coworkers found that the isostatic compositions in glassy alloys - predicted by rigidity theory - exists not just at a single threshold composition; rather, in many systems it spans a small, well-defined range of compositions intermediate to the flexible (under-constrained) and stressed-rigid (over-constrained) domains.^[11] This window of optimally constrained glasses is thus referred to as the intermediate phase or the reversibility window, as the glass formation is supposed to be reversible, with minimal hysteresis, inside the window.^[11] Its existence has been attributed to the glassy network consisting almost exclusively of a varying population of isostatic molecular structures.^[8][12] The existence of the intermediate phase remains a controversial, but stimulating topic in glass science.

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