Jamming and the glass transition
Introduction

Jamming occurs when a system develops a yield stress—behaves as a solid—in a disordered state. In practice, it is difficult to determine whether a system truly has a yield stress, or whether one has not yet waited long enough for the stress to relax. Thus, a more practical definition is that jamming occurs when the stress relaxation time of a system with no quenched disorder exceeds some fixed value, such as 103 to 105 seconds, in a disordered state.

Courtesy of D. J. Durian. Two gas-liquid foams. Surfactant (soap) molecules at the gas-liquid interface stabilize the gas bubbles against rupture. Foams like shaving cream (right) are typically made up of about 93% gas and 7% liquid. The foam on the left contains about 99% gas at the top, because the liquid has drained to the bottom due to gravity. Note that suspensions of hard spheres can only be packed to about 64% if the spheres are disordered; foams and emulsions can be packed to near 100% because the bubbles or droplets are deformable.

According to these definitions, many systems jam. Granular materials can flow when they are shaken or poured through a hopper, but jam when the shaking intensity or pouring rate is lowered. Similarly, foams and emulsions (dense colloidal suspensions of deformable gas bubbles or liquid droplets; see picture below) can flow when they are sheared, but are soft amorphous solids when the shear stress is lowered below the yield stress. These systems are athermal; random motions supplied by some driving force are necessary to induce any motion since thermal energy is insufficient to cause particle rearrangements.

In colloidal suspensions of smaller particles, such jamming also occurs when the pressure or density is sufficiently large. At low packing fractions, particles are free to diffuse and explore different configurations. As the packing fraction increases, however, the space available for motion decreases and the system becomes a disordered solid with a yield stress.

Even molecular systems jam as temperature is lowered—this is the glass transition. One of the oldest unsolved problems in the condensed phase is the nature of the glass transition. As temperature decreases, the stress relaxation time increases dramatically. Rapidly increasing relaxation times are not unique to jamming transitions; the relaxation time diverges near a second-order phase transition, as well. However, for such a transition, there is a structural length scale, the correlation length, which diverges. Because long length scale perturbations relax more slowly, the diverging correlation length gives rise to a diverging relaxation time. Near the glass transition, by contrast, it is not yet clear if there is any length scale that can be used to characterize the growth of the glassy response.

Is there really a meaningful connection among these different jamming transitions? At first glance the answer appears to be "No" because the phenomena have very different microscopic physical origins. In supercooled liquids, the increase in the relaxation time with decreasing temperature clearly has a thermal origin. In colloidal suspensions, increase in relaxation time with increasing density has an entropic origin. Finally, in athermal systems such as Bingham plastics (foams and emulsions) or granular materials, the increase in the relaxation time with decreasing applied stress appears to have a purely kinetic origin.

Despite these significant differences, however, there are striking similarities in the phenomenology of these different systems, summarized below.

These similarities of behavior do not prove that it is useful to tie these systems together by coining a new piece of jargon called "jamming." However, the similarities suggest that it is very important to establish how deep is the connection among these different systems. A common conceptual framework for jamming would be extremely powerful. Not only would it imply that ideas useful for understanding one system may be helpful to another system, but it would also imply that simple models may yield valuable insight.

The jamming "phase diagram"
"Jamming phase diagram" (A. J. Liu and S. R. Nagel, Nature 396, N6706, 21 (1998).) The jammed region, near the origin, is enclosed by the depicted surface.
The jamming phase diagram has been shown to be a good way of organizing experimental data. Here is the jamming phase diagram measured for a system of colloidal particles interacting by depletion attractions mediated by polymers. On the vertical axis, Trappe and coworkers varied the interaction strength U (rather than the temperature T) by varying the amount of added polymer in the system.

Different systems unjam as different parameters are varied. Glasses unjam as one raises the temperature, foams and emulsions unjam as one raises the shear stress, and colloidal glasses unjam as one lowers the packing density. These parameters are so different that it is difficult to see how one can compare the jamming transitions at a quantitative level. We pointed out that all three parameters are important to all systems, but the range over which they might be varied might be limited for a given system. In other words, these different parameters might be tied together by a "jamming phase diagram," shown to the right. The choice of axes is dictated by the parameters that control the transition to jamming in the different systems, namely temperature T, density φ, and shear stress S.

The ordinary state diagram for the glass transition would be in the (1/r)–T plane of the jamming phase diagram. At high density there is a transition between a supercooled liquid and a glass that occurs at Tg. As the density is lowered, Tg normally decreases. This glass-transition line is represented by the curve separating the jammed and unjammed regions in the (1/r)–T plane. There is, of course, considerable debate about whether the transition is a true thermodynamic one occurring at a nonzero temperature or whether the dynamics are completely arrested only at T=0. For the purposes of this discussion, the line separating the jammed (glassy) phase from the liquid state corresponds to a relaxation time that has increased to some fixed value, such as 103 seconds as is sometimes used to define Tg in supercooled liquids.

The ordinary phase diagram for a foam or emulsion would be in the (1/r)–σ plane of the jamming phase diagram. At fixed density, one must apply a shear stress higher than the yield stress in order for the system to flow. Thus, the yield stress as a function of density is the curve that separates the jammed and unjammed regions in this plane. As the density decreases towards close-packing, the yield stress decreases, as has been shown experimentally for emulsions and foams. We note again that it is impossible to tell whether the yield stress is truly nonzero or if it is only nonzero on the time scale of a rheological measurement. The same caveat that holds for the glass transition—that the transition corresponds to a relaxation time that has reached some fixed value—therefore holds for the entire surface of the jammed region.

Mode-coupling theorists suggested years ago that the colloidal glass transition and molecular glass transition are the same despite the fact that the control variables are different. The jamming phase diagram suggests a reason why the different jamming transitions might be related, independent of the accuracy of the mode-coupling approximation.

Point J in the jamming phase diagram

For systems with repulsive, finite-range interactions—that is, interactions that vanish beyond a certain distance—there is a special point on the jamming phase diagram, which we call point J.

Perhaps the most daunting problem in all jamming transitions is that jammed surface depicted above is typically not sharp, and is governed by when the system relaxation time exceeds experimental constraints. If a thermodynamic transition exists, it must lie underneath the jamming surface and is therefore inaccessible. This has made it very difficult to determine the nature of the jamming transition. We have shown that there is one point on the jamming phase diagram that is well-defined, namely, the point labeled J, along the 1/φ axis at zero temperature and shear stress. This point exists for systems with repulsive, finite-range potentials (systems with frictionless particles that do not interact beyond a certain distance, which defines their diameter). At low enough densities beyond Point J, none of the particles interact. At Point J, particles just come into contact, and at further compression particles interact and the pressure and zero-frequency shear modulus are nonzero. Point J has very special properties.

Remarkably, several models proposed for the glass transition, including the p-spin interaction spin glass, the mode-coupling approximation, a frustrated dimer model, and the Fredrickson-Andersen kinetically-constrained spin model have scaling exponents that, within mean-field theory, are the same as those we find at Point J. Many of these models exhibit dynamical slowing down and kinetic heterogeneities -- trademark features of the glass transition. This is a quantitative hint that jamming may be a universal phenomenon related to glassy behavior.

In summary, Point J is a very special point that may have important ramifications for the entire jamming surface. The properties of Point J are related to the mathematics of random close-packing geometries, the engineering of silos, the physical chemistry of colloidal suspensions and the physics of glass transitions. Understanding this point should lead to a better understanding of jamming in all of its manifestations.

Further reading:
Properties of the marginally jammed solid
Gedanken experiment for the measurement of thermal transport in jammed solids. We generate computer models of jammed packings and calculate their energy diffusivity, a spectral measure of transport that controls both sound attenuation and thermal conductivity.

Marginally jammed solids are all around us, from heaps of sand or gravel to glasses on the verge of liquefaction. By focusing on low-temperature at densities just above Point J, we discovered that the vibrational excitations of the marginally jammed state are completely different from that of most other materials. Subsequent theoretical progress was made in understanding these modes by considering the consequences of isostaticity at Point J. An isostatic system (precisely the right number of contacts to provide mechanical stability) is very special: isostaticity implies a dramatic change in the excitation spectrum of a solid compared to an ordinary crystal or isotropic elastic system - after all, many modes are on the verge of going soft. The new type of excitation has important implications for energy transport. Moreover, understanding their genesis and structure can have important implications for understanding glassy materials generally. In particular, vestiges of marginally-jammed behavior are manifested in the existence of the boson peak, believed to be responsible for many of the characteristic low-temperature phenomena in glasses.

Further reading: See also:
Effective temperatures in sheared systems near jamming
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Movie of a steady shear simulation. Double click to start. More movies

One of the outstanding challenges in condensed matter is to understand the collective, many-body behavior of systems far from equilibrium in terms of their microscopic properties.

When a liquid is quenched through the glass transition temperature T0, it falls out of equilibrium and becomes a glass. Below T0, the thermal energy is insufficient for the system to surmount energy barriers on accessible time scales, so the glass explores relatively few configurations. However, if the glass is held in contact with a thermal reservoir and sheared at a fixed rate, it can reach a steady state in which it explores many different minima in the energy landscape, even if the temperature of the reservoir is well below T0. The steadily sheared glass is far from equilibrium: energy is continually supplied on long time and length scales via the boundaries, and is removed on short scales by the thermal reservoir. Nonetheless, simulations show that fluctuations in such systems are well described by an effective temperature Teff that is higher than the bath temperature, as predicted theoretically by Cugliandolo and Kurchan. Nine different definitions yield a common value of Teff, providing strong numerical evidence for the utility of the concept.

However, if we want to understand the behavior of a material, measuring the temperature is only a start. Now that effective temperatures have been shown to describe steady-state sheared glassy systems remarkably well, it is past time to explore the possible consequences of the effective temperature for materials properties. Our recent work shows that there in the regime where shear-induced fluctuations dominate over thermal fluctuations, the behavior of the system appears to be controlled by the effective temperature. Thus, Teff appears to play an important role for determining materials properties. Further reading:

See also:
A Kinetically-Constrained Model Explains Dynamic Heterogeneity in Glass and Granulates

Glass-forming liquids, granular systems, colloids, emulsions and foams exhibit a huge increase in relaxation time while the structure is hardly affected, thus a growing correlation length remains elusive. One of the leading suggestions for a relevant length scale comes from the heterogeneity of the dynamics. Granular and colloidal experiments have indeed demonstrated dramatic increase in dynamic correlation lengths as jamming in approached. However, in molecular glass formers a significant increase in the dynamic correlation length is typically followed by a range over which the relaxation time may grow by more than ten orders of magnitude while the spatial extent of heterogeneity in the dynamics increases by less than a factor of two. In this paper we resolve this puzzle using a kinetically-constrained model. Our tractable lattice model includes physical mechanisms that separately mimic the effects of density, temperature and non-equilibrium driving. Here, the ingredient responsible for slow dynamics is that the ability of a particle to hop between adjacent lattice sites depends on the occupation of neighboring sites in a manner that models the geometric setting constraining allowed moves in particulate systems. We separate the effects of density, temperature and driving and show that jamming resulting from increasing density gives rise to dynamic heterogeneity that grows unboundedly. Whereas decreasing temperature or driving eventually leads to a saturation of the dynamic correlation length even though the relaxation time diverges.

Further reading:
Finite-Size Scaling

It has been argued that jamming is a phase transition because it displays a finite jump in the number of contacts per particle, power law scalings in both the excess contact number and the shear modulus, and diverging length scales. At the heart of the theory of phase transitions is scaling collapse -- the idea that measurements in finite systems can be collapsed onto a single curve by scaling variables by the size of the system. While many quantities, such as the contact number and the shear modulus are independent of system size, we have recently shown that finite size effects exist very close to the transition. Furthermore, these effects collapse with system size, confirming that jamming displays the essential characteristics of a phase transition.

Further reading:
The Rigidity Length Scale

Much of our theoretical understanding of the vibrational properties of systems near the jamming transition comes from a length scale called l*, originally proposed by Wyart et al. (EPL 72, 486 (2005)). We have identified the origin of this length scale as the minimum size that a finite system must have in order to be rigid. As the movie below shows, a large system with free boundary conditions is dominated by a single rigid cluster (particles in red). However, when the size of the system decreases sufficiently, the rigid cluster suddenly breaks up. The size of this minimal rigid cluster is l*. We have now formalized this notion quantitatively and checked it numerically.

Further reading:
Generalized Stability

While numerical simulations are inherently limited to finite sizes, periodic boundary conditions may be employed to estimate bulk properties of a system. However, it was recently shown by Dagois-Bohy et al. (PRL 109, 095703 (2012)) that systems prepared in a square, periodic unit cell can be unstable to shear. These instabilities are associated with the shape of the unit cell. We generalize this notion by considering arbitrary deformations throughout the first Brillouin Zone. We show that these instabilities, which vanish at large system sizes and at high pressures, can be understood through plane waves and the behavior at ω*.

Further reading: