Many common materials such as glass, compacted sand and toothpaste have a solid’s rigidity but a liquid’s disordered microscopic structure. Despite their diversity, however, such “amorphous solids” share many mechanical properties. Now, Eric DeGiuli of the École Normale Supérieure in Paris has devised a field theory inspired by particle and condensed-matter physics that he says can describe how any amorphous solid behaves when subject to stress.
The atoms and molecules within amorphous solids are locked together in space but lack the uniform crystalline structure of conventional solids. This non-crystalline arrangement confers many useful properties, including the ability to withstand large forces. Many modern buildings, for example, are built using pre-stressed concrete, while some mobile phones are made from metallic glass – an amorphous solid that has many metallic properties, such as electrical conduction, but which is much stronger than a metal.
However, when glass and other amorphous solids are subject to shear stress they undergo an “avalanche” effect. A certain level of stress makes one portion of the material unstable, which then leads to instability on another portion, and so on. This makes it hard to predict at what level of stress an amorphous material will break up, so engineers often have to build in large and costly safety margins.
Invaluable theory
Scientists have observed this avalanching by compressing nanometre-high pillars until they collapse. But extrapolating those results up to macroscopic sizes is “quite challenging”, says DeGiuli given the need to consider materials with a wide range of shapes and composition. This, he says, makes a good theory invaluable. “Theory can guide you to make materials with specific properties,” he says.
That is easier said than done. Thanks to their regular crystalline structure, normal solids exist at a global energy minimum and their mechanical properties are therefore relatively easy to predict. Amorphous solids instead sit in what are known as metastable states. When molten silicate is cooled down very quickly it is not able to reach equilibrium and gets stuck in a local energy minimum, so becoming glass. Without any external forcing the material will retain that state forever – its molecules remaining in fixed relative positions. DeGiuli says that developing a theory that can predict how these different metastable states are “sampled” has proved a headache.
One approach is to assume that each small volume within an amorphous solid responds elastically when the material is subject to stress. This might explain how stress fields in glass behave. But this does not seem to apply to sand – experiments showing that grains of sand do not interact with one another elastically when the material is compacted.
Isotropic equilibrium
In his latest research, DeGiuli instead relies on two very straightforward assumptions. One, that amorphous solids are essentially isotropic – the same in all directions. And two, that such materials are in mechanical equilibrium, which means that each small volume within them experiences no net force or torque.
DeGiuli is not the first person to work from these assumptions. In 2009, Bulbul Chakraborty and Silke Henkes of Brandeis University in the US built on a statistical-mechanics model developed by Sam Edwards at the University of Cambridge in the 1980s in order to describe how stress is distributed within granular solids such as compacted sand. That model assumed that a material is equally likely to end up in any of the metastable states available to it.
To avoid the difficulty of defining and then counting metastable states, DeGiuli instead takes a more macroscopic approach. In a paper in Physical Review Letters, DeGiuli describes how he used renormalization, a mathematical scheme that reveals how the behaviour of a physical system depends on the scale used to view that system. First used in particle physics in the 1970s, renormalization has been adapted by DeGiuli to analyse a non-equilibrium system for the first time. In doing so, he could characterize a potentially infinite number of metastable states by just a handful of mathematical terms by probing materials at scales far bigger than the size their constituent particles.
Mountainous landscape
To explain his approach, DeGiuli makes an analogy between an amorphous solid and rainfall hitting a mountainous landscape – the idea being to work out where the water will end up. “I was able to construct a theory that only considers the special parts of the phase space – the rivers and lakes – rather than every point on the landscape,” he says.
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DeGiuli’s field theory is more general than that of Chakraborty and Henkes, being applicable to all amorphous solids rather than just granular solids. It also predicts how stress is correlated across materials in three dimensions (rather than being limited to two). To put his theory to the test, he now plans to use it to model the mechanical behaviour of glass at low temperatures and see if he can predict certain anomalous-looking results seen in experimental data.
Not everyone is convinced by the latest research, however. Anaël Lemaître of the Laboratoire Navier in Paris, who has demonstrated the existence of long-range stress correlations in amorphous solids, argues that DeGiuli’s analysis is not rigorous because it is still based on Edwards’ model. “Sam Edwards was very careful when he introduced his theory that it was just based on an ‘analogy’ with statistical mechanics,” he says. “It may provide insights but cannot support a general proof on the properties of amorphous solids.”
