# Rigidity theory (physics)

Rigidity theory, or topological constraint theory, is a tool for predicting properties of complex networks (such as glasses) based on their composition. It was introduced by Phillips in 1979 and 1981, and refined by Thorpe in 1983. Inspired by the study of the stability of mechanical trusses as pioneered by James Clerk Maxwell, and by the seminal work on glass structure done by William Houlder Zachariasen, this theory reduces complex molecular networks to nodes (atoms, molecules, proteins, etc.) constrained by rods (chemical constraints), thus filtering out microscopic details that ultimately don't affect macroscopic properties. An equivalent theory was developed by P.K. Gupta A.R. Cooper in 1990, where rather than nodes representing atoms, they represented unit polytopes. An example of this would be the SiO tetrahedra in pure glassy silica. This style of analysis has applications in biology and chemistry, such as understanding adaptability in protein-protein interaction networks. Rigidity theory applied to the molecular networks arising from phenotypical expression of certain diseases may provide insights regarding their structure and function.

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 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. Typically, the conditions for glass formation will be optimal if the network is isostatic, which is for example the case for pure silica. Flexible systems show internal degrees of freedom, called floppy modes, 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.

### Derivation of isostatic condition

The conditions for isostaticity can be derived by looking at the internal degrees of freedom of a general 3D network. For $N$ nodes, $N_{c}$ constraints, and $M_{eq}$ equations of equilibrium, the number of degrees of freedom is

$F=3N-N_{c}-M_{eq}$ The node term picks up a factor of 3 due to there being transnational degrees of freedom in the x, y, and z directions. By similar reasoning, $M_{eq}=6$ in 3D, as there is one equation of equilibrium for translational and rotational modes in each dimension. This yields

$F=3N-N_{c}-6$ This can be applied to each node in the system by normalizing by the number of nodes

$f=3-n_{c}$ where $f={\frac {F}{N}}$ , $n_{c}={\frac {N_{c}}{N}}$ , and the last term has been dropped since for atomistic systems $6< . Isostatic conditions are achieved when $f=0$ , yielding the number of constraints per atom in the isostatic condition of $n_{c}=3$ .

### Developments in glass science

Rigidity theory allows the prediction of optimal isostatic compositions, as well as the composition dependence of glass properties, by a simple enumeration of constraints. These glass properties include, but are not limited to, elastic modulus, shear modulus, bulk modulus, density, Poisson's ratio, coefficient of thermal expansion, hardness, and toughness. In some systems, due to the difficulty of directly enumerating constraints by hand and knowing all system information a priori, the theory is often employed in conjunction with computational methods in materials science such as molecular dynamics (MD). Notably, the theory played a major role in the development of Gorilla Glass 3. Extended to glasses at finite temperature and finite pressure, rigidity theory has been used to predict glass transition temperature, viscosity and mechanical properties. It was also applied to granular materials and proteins.

In 2001, Boolchand and coworkers found that the isostatic compositions in glassy alloys - predicted by rigidity theory - exist 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. 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. Its existence has been attributed to the glassy network consisting almost exclusively of a varying population of isostatic molecular structures. The existence of the intermediate phase remains a controversial, but stimulating topic in glass science.