A comparison of jamming behavior in systems composed of dimer- and ellipse-shaped particles

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Carl F. Schreck, Ning Xu and Corey S. O'Hern

Soft Matter 6 (2010) 2960-2969


wiki entry by Emily Russell, Fall 2010


The article can be found here.


Overview and Comments

This paper touches on several properties of jammed systems of concave dimers and convex elliptical particles, including jamming volume fraction, average contact number at jamming, vibrational mode spectrum, variation of shear modulus with volume fraction, stress-strain relations and yield stresses, and nematic ordering. It is a somewhat dense paper that goes rather quickly through all of these properties, but it argues effectively that the details of the shape of anisotropic particles can have significant effects on the behavior of jammed systems. Dimers behave similarly to simple disks in many ways, whereas ellipses show novel behaviors.


Simulations

Fig. 1 Definition of the aspect ratio <math>\alpha</math> = a/b (ratio of the major to minor axes) for (a) ellipses and (b) dimers.

The authors consider the packing of two geometrical shapes, as shown in Fig. 1: ellipses, and dimers made up of two identical circular disks. In each case, they consider aspect ratios <math>\alpha</math> in the range <math>1\leq\alpha\leq2</math>. All simulations were done in two dimensions, so that each particle has three degrees of freedom, two translational and one rotational. Simulations were done on bidisperse systems, with populations of both large and small particles with the same aspect ratio, to inhibit crystallization so that jamming, glassy behavior could be studied.

The particles were modeled as interacting with a soft repulsive potential (which required some tricky geometry for working out the contact distance of two randomly-oriented ellipses). The packings were generated by randomly placing particles at a low volume fraction, and then incrementally increasing their size and relaxing the system until a jammed system was reached. Shear experiments were simulated quasi-statically, introducing an incremental shear, relaxing the system and measuring the strain, and repeating until a final shear strain of 1 was reached.


Results

Fig. 2 Ensemble averaged contact number <math>z_J</math> at jamming as a function of aspect ratio <math>\alpha</math> for dimers (squares) and ellipses (circles) for N = 480 particles.

The simplest result found was a difference in the average contact number between ellipses and dimers. Jammed packings of circular disks have an average contact number of <math>\approx 2d</math>, where d is the dimension (here d=2); each disk is constrained in every degree of freedom. The results for the dimers were similar; the average contact number was 6 = 2<math>d_f</math> (the third degree of freedom is rotational). The ellipses, however, showed a very different behavior, with the contact number rising from 4 to 6 as aspect ratio increased (Fig. 2). The case of <math>z_J < 2d_f</math> is referred to as hypostatic.

New classes of vibrational modes also appeared in these systems. The high-frequency modes were mainly translational, similar to the modes in a system of disks. At lower frequencies, the modes were dominated by rotations, not surprising given the broken rotational symmetry. What was more remarkable was the discovery in the ellipse system, but not in the dimer system, of mainly rotational modes in which the energy changed quartically with amplitude, rather than quadratically, above the very smallest displacements.

The authors also considered several aspects of the behavior of the systems under shear. To summarize all of their findings here would be nearly to duplicate the paper; however, perhaps the most interesting finding in this category was the change in scaling of the static linear shear modulus with volume fraction. In a system of jammed disks, G scales as <math>(\phi - \phi_J)^{0.5}</math>. This same scaling relation was found for the dimers; however, in the ellipse system, the scaling exponent found was one. The authors refer to previous work which showed that this change in exponent was related to the presence of the quartic vibrational modes mentioned above.

A final observation made, which although made briefly was quite interesting, was on the nematic ordering under shear. This was a case in which both ellipses and disks behaved similarly (and which can of course not be compared to disk packings); the nematic order increases with strain to a plateau value which depends on the aspect ratio.


Discussion

The similarity between dimer packings and disk packings does not seem so remarkable given that the dimers are in fact made up of 'fused' disks. Indeed, the authors mention in their discussion on "Future Directions" that they hypothesize 'peanut'-shaped particles, convex particles with a large radius of curvature at their narrowest point, may be hypostatic as the ellipses were found to be. To get real insights into the effects of geometry on jamming properties, a wider range of geometries should be considered. I am also curious how much the bidispersity affected the results.

This also seems like an area readily open to experimentation. Progress has been made in recent years on synthesizing dimer-shaped colloids, and studies of these systems will surely turn up some interesting phenomena. I have not heard of experimental studies on ellipses, but it seems like a plausible experiment to perform, on a ball-bearing scale if not yet on a colloidal scale.

This paper was somewhat dense in places, but made a convincing argument that many properties of a jammed system differ significantly depending on the detailed morphology of the particles, an observation which is not immediately intuitively obvious.