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 James Charles Phillips in 1979 and 1981, and refined by Michael 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. The same condition  applies to random packing of spheres, which are isostatic at the jamming point. 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 formula_1 nodes, formula_2 constraints , and formula_3 equations of equilibrium, the number of degrees of freedom is The node term picks up a factor of 3 due to there being translational degrees of freedom in the "x", "y", and "z" directions. By similar reasoning, formula_5 in 3D, as there is one equation of equilibrium for translational and rotational modes in each dimension. This yields This can be applied to each node in the system by normalizing by the number of nodes where formula_8, formula_9, and the last term has been dropped since for atomistic systems formula_10. Isostatic conditions are achieved when formula_11, yielding the number  of constraints per atom in the isostatic condition of formula_12. An alternative derivation is based on analyzing the shear modulus formula_13 of the 3D network or solid structure. The isostatic condition, which represents the limit of mechanical stability, is equivalent to setting formula_14 in a microscopic theory of elasticity that provides formula_13 as a function of the internal coordination number of nodes and of the number of degrees of freedom. The problem has been solved by Alessio Zaccone and E. Scossa-Romano in 2011, who derived the analytical formula for the shear modulus of a 3D network of central-force  springs (bond-stretching constraints): formula_16. Here, formula_17 is the spring constant, formula_18 is the distance between two nearest-neighbor nodes, formula_19 the average coordination number of the network (note that here formula_20 and formula_21), and formula_22 in 3D. A similar formula has been derived for 2D networks where the prefactor is formula_23 instead of formula_24. Hence, based on the Zaccone–Scossa-Romano expression for formula_13, upon setting formula_26, one obtains formula_27, or equivalently in different notation, formula_28, which defines the Maxwell isostatic condition. A similar analysis can be done for 3D networks with bond-bending interactions (on top of bond -stretching), which leads to the isostatic condition formula_29, with a lower threshold due to the angular constraints imposed by bond-bending. 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 the context of soft glasses, rigidity theory has been used by Alessio Zaccone and Eugene Terentjev to predict the glass transition temperature of polymers and to provide a molecular-level derivation and interpretation of the Flory–Fox equation. The Zaccone –Terentjev theory also provides an expression for the shear modulus of glassy polymers as a function of temperature which is in quantitative agreement with experimental data, and is able to describe the many orders of magnitude drop of the shear modulus upon approaching the glass transition from below. 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.