Most soft materials, such as sand, can be in either a solid-like or a liquid-like state. New experiments probe the surprisingly rich nonlinear physics that can occur in between these two states. See Letter p.355
All around us, things are falling apart. The foam on our cappuccinos looks solid, but gentle stirring irreversibly changes its shape. Sand, a granular material, mimics a solid when we walk on the beach but a liquid when we pour it out of our shoes. Such examples suggest that we can think of the mechanics of soft disordered materials as either jammed (solid-like) or unjammed (freely flowing). On page 355 of this issue, however, Bi et al.1 describe experiments showing that such materials come in more than just these two flavours.
The hybrid behaviour of sand and foams has fascinated physicists for decades, and lies at the heart of Liu and Nagel's jamming diagram2 (see Fig. 1a of the paper1). The idea behind this diagram is that the mechanical state of a whole range of soft materials — such as granular media, pastes, foams and emulsions — is controlled by how densely their constituents (grains, bubbles or droplets) are packed. Dense packings are jammed, loose packings are unjammed, and when the constituent particles just touch, the material is said to be marginal. Think of mayonnaise, an emulsion of oil droplets in water: only when a sufficient amount of oil is added to the mixture do the oil droplets start to touch and the mayonnaise acquires its solid-like consistency.
Bi and colleagues' experiments1 suggest that the jamming diagram should be revisited to describe a broader variety of states than the strictly jammed and unjammed ones. The authors placed a granular material consisting of small plastic discs in a box whose side walls could be moved in order to compress or shear (deform without compression) the material. The discs were photoelastic and so permitted direct visualization of the forces operating between them when the material was subjected to deformation: the more incident light that went through the material, the larger the forces3.
The first question Bi et al. addressed was, what happens when such granular material is sheared? Typically, when we spread mayonnaise on a sandwich, smear foam on our skin or kick a sandcastle, we unjam these materials. By contrast, the authors1 find that, when sheared under constant volume, collections of loose, unjammed discs build up pressure and resist further deformation — they become rigid. This phenomenon is reminiscent of Reynolds' dilatancy, in which granular materials expand when sheared under constant pressure. A familiar manifestation of this is how footprints on a wet beach tend to become dry as the deformed sand expands and sucks in water.
The second question Bi et al. tackled was, what is the nature of the novel states in which mechanical rigidity is generated by shearing? The authors show that these states are anisotropic, as evidenced by the patterns formed by the bright, load-bearing discs. For moderate anisotropies, the states seem to be fully jammed: they are rigid enough to resist deformation in any direction. These are shear-jammed states. However, in the case of strong anisotropy, the packings become fragile: the discs come into contact with each other mostly in one direction (Fig. 1). Such fragile packings4 can resist deformation in some directions but not in others. These packings are thus neither fully jammed nor fully unjammed.
To account for their findings, Bi et al. suggest a modification to the jamming diagram: in addition to the areas describing the jammed and unjammed states, there should be an in-between region for the fragile and shear-jammed states (see Fig. 1b of the paper1). All of these states should meet near a point known as the marginal or jamming point, at which the rigidity vanishes.
This proposed diagram is specifically geared towards frictional granular media. But the idea of fragile states5, which are extremely susceptible to perturbation and emerge from the marginal point, is more general. Many soft materials — including granular media, emulsions, foams and polymer networks — can be prepared in a state of vanishing rigidity and thus become marginal materials. Several examples are emerging6,7,8,9,10 that show precisely how the mechanical response of such marginal materials becomes strongly nonlinear. The closer they come to the marginal point, the lower the driving force needed to access this fully nonlinear regime. At the marginal point, even the tiniest perturbation produces an extremely nonlinear mechanical response.
One example of the nonlinear behaviour of fragile matter is the mechanics of networks of randomly arranged springs. These reach a marginal state when the average number of springs equals twice the number of network nodes. Such networks are floppy if fewer springs are present and become rigid when extra springs are added, provided that they are under relatively weak stress. Between these two regimes, there is a window of extreme response: when the spring network is near its marginal state, its response becomes completely nonlinear6.
A second example is the flow of foams and soft colloidal suspensions. Under low stress, these exhibit a liquid-like state for low densities and a solid-like state for high densities. In between these two states, an intermediate regime of nonlinear flow arises7,8.
A third example is waves in granular media. Under large pressure, linear elastic waves occur. However, at low pressures the material approaches its marginal point, and even a gentle touch can generate strongly nonlinear shock waves9,10.
Bi and colleagues' work1 underscores the surprisingly rich nonlinear physics that governs marginal soft matter. The simplest manifestation of this nonlinear behaviour can be found near the marginal point. In many cases, it is unclear exactly what happens when such fragile materials are subjected to stress. But what is clear is that being marginal matters.
Bi, D., Zhang, J., Chakraborty, B. & Behringer, R. P. Nature 480, 355–358 (2011).
Liu, A. J. & Nagel, S. R. Nature 396, 21–22 (1998).
Majmudar, T. S. & Behringer, R. P. Nature 435, 1079–1082 (2005).
Cates, M. E., Wittmer, J. P., Bouchaud, J.-P. & Claudin, P. Phys. Rev. Lett. 81, 1841–1844 (1998).
de Gennes, P.-G. & Badoz, J. Fragile Objects: Soft Matter, Hard Science, and the Thrill of Discovery (Springer, 1996).
Wyart, M. et al. Phys. Rev. Lett. 101, 215501 (2008).
Olsson, P. & Teitel, S. Phys. Rev. Lett. 99, 178001 (2007).
Nordstrom, K. N. et al. Phys. Rev. Lett. 105, 175701 (2010).
Nesterenko, V. F. Dynamics of Heterogeneous Materials (Springer, 2011).
Gómez, L. R., Turner, A. M., van Hecke, M. & Vitelli, V. Phys. Rev. Lett. (in the press); Preprint at http://arxiv.org/abs/1108.5688 (2011).
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