Critically jammed

Diverse forms of matter around us can simply be classified as those that flow and those that don’t and exhibit rigidity, a resistance to deform when a force is applied. This elementary distinction informs our ability to comprehend their properties and to manipulate them. It is easy enough to pick up a spoon, a solid that pushes back against our fingers, and pour a spoonful of sugar into and stir a cup of coffee, a liquid that will flow in response to the force we exert. A cupful of coffee, sugar poured into a cup, and water flowing down an incline, are examples of flowing matter, whereas a spoon or a lump of rock or window glass are solids, characterized by their rigidity. Of course, the same substance can exist in either a flowing or rigid state. Water that runs off a roof with ease on a warm day will turn rigid when the temperature drops and will hang down the edges as icicles, resisting gravity. Somewhat less obviously, one may think of the sugar that pours off the spoon easily as being in a rigid state inside the jar, as may be noticed if one attempts to press it down with the back of a spoon. These transformations are of obvious importance in understanding the properties of these substances. Of these, the transformation of water into ice is a well-understood, so-called first-order phase transition, involving discontinuous changes in properties, such as density. The rigidity of ice is understood in terms of the periodic arrangement of molecules, which reduces—or breaks—the continuous translational symmetry of the liquid’s microscopic structure, in which molecules have no such regularity. But the emergence of rigidity in substances that remain disordered when they transform from flowing to rigid states (such as sugar compacted into a jar) remains mysterious, and many fundamental aspects of the transition to the rigid state are poorly understood. Goodrich et al. (1), in work reported in PNAS, propose a way forward in understanding such “jamming” transitions.

The Jamming Transition

Transitions from flowing to rigid, disordered states arise in many contexts. One example is the glass transition, when a liquid is cooled rapidly, so that the disorder in its microscopic structure remains in the amorphous solid form at low temperatures. This is how window glass is made. The occurrence of jamming in granular matter, which Goodrich et al. (1) address, is a somewhat different case. In glasses like silica, the constituent particles, whose assemblies one aims to understand, are molecules, subject to incessant thermal motion. In granular matter, like a spoonful of sugar whose grains are millimeters in size, the constituent particles are macroscopic and thermal motion is insignificantly small. Thus, granular matter is described as being athermal, whose flow behavior depends on density, applied stresses, and so forth, but not temperature (or, for that matter, the internal structure and composition of individual grains). The rigidity of granular assemblies depends largely on how they pack, subject to interactions that mainly serve to keep them out of each other’s way (although other interactions are important in some cases). Granular matter is ubiquitous, ranging from the commonplace to the esoteric: sugar, ground coffee, all kinds of food grains, powders, sand dunes, icebergs in ice floes, rock avalanches, asteroid belts, the rings of Saturn, intergalactic dust, and so forth. It is claimed that granular matter constitutes the second-most manipulated material in industry, next only to water (2). Depending on the case, the size of the individual grains can vary widely in size. In addition to mostly hard grains in the examples above, foams and emulsions are important instances. Viewed as athermal assemblies of nearly frictionless, deformable spheres, they are closer in some respects to theoretical idealizations.

The key variable that induces the transition from flowing to jammed states in granular matter is density. Idealized models for studying this transition have been assemblies of hard or soft deformable spheres, with or without frictional interactions. Although regular or crystalline packings exist for these packings and are rigid, much attention has focused on the transition from a flowing to a rigid state when packings remain disordered as density increases. The nature of this transition, studied extensively for frictionless soft spheres, is a curious mixture compared with conventional first-order and second-order (or so-called continuous) transitions. Importantly, the transition shares many features of continuous phase transitions. The transformation of an undifferentiated fluid at high temperatures to distinguishable liquid and gas phases is a well-studied example of a continuous transition. The critical point (specific values of temperature and pressure at which liquid and gas phases come into existence) of such a transition is characterized by continuous changes in some properties like density, but has the singular behavior of quantities like compressibility, which measure how strongly the density may vary with a change in applied pressure.

An important characteristic of a continuous transition is that many quantities exhibit power law dependences on the distance to the critical point. Thus, the compressibility becomes arbitrarily large, or diverges, as the critical point is approached along a path of fixed, critical, density ρ_c, as κ_T ∼ (T – T_c)^−γ, and the density of the liquid ρ_L below the critical point varies as (ρ_L − ρ_c) ∼ (T_c – T)^β. Exponents such as β and γ are termed critical exponents. Given that the critical point can be approached in different ways (varying the temperature or pressure, for example), and that many quantities exhibit power law scaling, there are many such critical exponents. Curiously, these exponents appear to be interrelated and not all independent. Understanding these relations through scaling relationships between different relevant variables is among the triumphs in the study of critical phenomena, which gave rise to potent and widely used ideas, such as scaling, scale invariance, universality, and renormalization. The scaling hypothesis, resulting from the work of Benjamin Widom (3) and others, follows the observation that the free energy (whose derivatives lead to various thermodynamic functions of interest) depends on variables like temperature and density through a function of specific power law ratios of these variables. The scaling functions involved imply the observed relationships between critical exponents.

Scaling Analysis Near the Jamming Point

What Goodrich et al. (1) propose is a scaling function that describes the jamming transition. Leading up to this work are a series of investigations, many through numerical simulations of soft sphere packings (e.g., ref. 4), under athermal conditions that reveal that in many ways the jamming transition indeed displays characteristics of a critical point (5). Approaching the jamming density ϕ_J from higher values, many quantities vanish as power laws in (ϕ − ϕ_J), which include the pressure, the bulk and shear moduli, and the excess in the number of contacts per sphere Z over the isostatic value Z_c, which is the minimum number of contacts needed to hold each sphere in place. The isostatic contact number has been estimated by a counting argument that equates the total number of degrees-of-freedom of the spheres to the number of constraints to their movement arising from contact with other spheres. In the absence of frictional interactions, the isostatic contact number, estimated this way, and indeed observed, is Z_c = 2d, where d is the spatial dimension (mostly, d = 2, 3 in these studies). It has also been argued, and verified in a number of ways (6–8), that the vanishing of rigidity approaching the jamming density is also accompanied by a length scale that grows and diverges at the jamming point. Although the jamming density itself appears not to be unique (9), the scaling with respect to the jamming density is robust. Alternatively, one might think of a parameter such as ΔZ = (Z − Z_c) as the relevant variable with respect to which one analyzes the power law forms.

These observations involve a relatively large number of exponents, and given the appearance of a critical phenomenon, beg the question of how the interrelations between these exponents may be pursued, through an analysis of scaling relations between relevant quantities. Some efforts have indeed been made to find scaling relations, considering also the finite sizes of studied systems (10, 11). But so far, a full-fledged scaling theory that aims to bring together scaling relationships among all of the relevant control parameters, and attempts to generate a scheme with which to explore the set of exponent relationships, has arguably not been proposed. This is what Goodrich et al. (1) present in their paper, by introducing a scaling function for the elastic energy in terms of scaling variables involving density, contact number, shear strain, and the finite size of the studied system. Because the pressure and shear stress are obtainable as derivatives of the elastic energy, and the bulk and shear moduli as

The work of Goodrich et al. constitutes a concrete and exciting step forward in unraveling the mysteries of the jamming transition.

second derivatives, one obtains four exponent relations. Goodrich et al. (1) show that a fifth relationship can be derived between exponents describing pressure and shear stress. This approach reduces the number of independent exponents to three, for the values of which, the authors point out, theoretical arguments exist (12, 13). The effectiveness of the equilibrium-like description of the jamming transition, nonequilibrium in character and subject to protocol dependences, is remarkable, and must depend on the right choice of variables to describe the constraints on the packings. Goodrich et al. (1) describe the excess contact number ΔZ as a natural choice, and use it in analyzing corrections to scaling at large contact numbers. Because ΔZ has centrally been discussed in relation to a diverging length scale upon approaching the jamming transition, a physically appealing picture of the emerging scale invariance at the jamming point may be at hand. This is overall a very tidy picture, and a step forward in understanding the jamming transition along well-trodden paths in comprehending critical phenomena.

There are some obvious questions as to how this analysis can proceed. Although not central to the present analysis, shear strain is among the control parameters of the theory developed, and it will be interesting to see how the current analysis (1) may be extended to understanding shear jamming that arises in sheared packings below the jamming transition (14–16), and possible criticality that may be involved. Another important question concerns the extension of this approach to densities below the jamming point. Elastic energy cannot provide a starting point there because it is uniformly zero in that regime, and suitable extensions will need to be conjured. There has been considerable recent activity in understanding the glass transition on the basis of rigorous analysis for infinite dimensional systems, and in particular to make contact from that perspective with the jamming transition (e.g., ref. 17), approached from lower densities. An encouraging observation in this regard is that the shear modulus computed using the same formulation agrees with the observed scaling near the jamming point (18). Making contact with such analyses also will involve a more unified description of the glass transition and jamming, a holy grail set up in the formulation of the jamming phase diagram (5), which envisages a jamming surface separating jammed states from unjammed states obtained by increasing volume, temperature, or applied stress. Aspects of jamming have also been explored recently in biological assemblies [e.g., yeast populations (19) and biological tissue (20)], with or without self-propelled or active dynamics playing an important role, and analogies of scaling behavior seen near the jamming point have already been analyzed (20). The jamming transition that Goodrich et al. (1) apply their scaling theory to is a nonequilibrium situation, a step away from conventional critical phenomena. Can it be taken a step further, to include active and biological assembly? These and related open questions will no doubt keep investigators busy for some time to come, but the work of Goodrich et al. constitutes a concrete and exciting step forward in unraveling the mysteries of the jamming transition.