The packing of granular polymer chains
Original entry: Sujit S. Datta, APPHY 225, Fall 2009.
L. N. Zhou, X. Cheng, M. L. Rivers, H. M. Jaeger and S. R. Nagel, Science 326, 408 (2009).
A major thrust of soft matter physics research in the past decade has been to elucidate the nature of "jamming", the manner in which various soft materials transition to a state in which they are rigid and resistant to shear. This is generally thought to be tuned by shear stress, temperature, or packing density, in a manner that is thought to be common to a wide variety of materials (colloidal suspensions, powders and grains, or polymer melts, for example). Random packings of athermal spheres, for example, can be rigid and resistant to shear at large enough packing densities. Polymer melts, on the other hand, can also quickly transition to a jammed or "glassy" state at sufficiently low temperatures. While both of these transitions have been thoroughly explored independently, very little work draws explicit connections between the two. This paper reports on experiments studying the packing of flexible granular chains (structurally similar to polymers), and the manner in which they jam. Interestingly, this transition seems to have much in common with the polymeric glass transition, thus suggesting universal behavior in the manner in which granular systems and polymer melts "jam".
Conceptually, Zhou et al's experiments are simple. They made millimeter-sized chains of metal balls, packing them into a mechanical shaker and compacting them with many vibrations -- this structure is by construction mechanically stable, and thus "jammed". Interestingly, they find that as the chain length increases, the final packing density decreases to an asymptotic value; this is slightly larger for floppier chains, consistent with intuition (in the limit of infinite floppiness, the packing density should be the same as the maximum packing density of ~64% for hard spheres, independent of the chain length). But how are long, floppy chains of balls mechanically stable at packing densities much smaller than that corresponding to mechanically stable hard sphere packings? That is to say, what confers stability to a packing of granular chains?
Using x-ray tomography, Zhou et al were able to map out the 3-dimensional structure of their granular polymer packings. By studying this structure, they found that long floppy chains tend to form small loops along their length; these are obviously more rigid than the overall chain. It is the jamming of these small loops that confers mechanical stability to the overall packing. Not every single ball along the length of a loop needs to be locally jammed for the overall loop to be in a jammed configuration -- this is why the jamming of loops is able to support stresses at packing densities much "looser" than random close packing of hard spheres.
How is this related to the polymeric glass transition -- the transition of a polymer melt from a liquid-like to solid-like state as temperature is decreased below a given glass transition temperature Tg? As described by Flory in 1950, Tg is inversely related to the molecular weight M of polymer; as M diverges, Tg increases to a given constant value. This is similar to Zhou et al's observations of the increase in the reciprocal of the jamming packing density (the analog of Tg) of granular chains as the chain length (the analog of M) diverges. Thus, this work strengthens the possible connection between jamming of granular materials and the polymeric glass transition.