59th Annual Report on Research 2014

            The effects of pinning in solids are critical to many problems, including superconductors and charge density waves.  Our work focuses on the effect of the presence of pinned particles on jammed solids. At the time of our first annual report to the ACS PRF, good progress was made in terms of writing codes and producing preliminary data on systems of soft spheres in 2d near "Point J", the zero-temperature, jamming phase transition. Our work in the second grant year focused on completing one of the projects from our first year. This was the universal behavior of jamming in the limit of a vanishing density of pinned particles. In continued collaboration with the Liu group in the Physics Dept. at the University of Pennsylvania, we completed several million simulations on a High Performance Computing (HPC) cluster by the end of Spring, 2014.   

            Figure 1 is representative of our systems of interest:  N_f = 1, 2, 3 and 4 obstacles, whose disordered positions were fixed before the remaining "mobile" particles were allowed to relax to their lowest-energy configuration at zero-temperature.  Data were obtained from systems with numbers of particles ranging from 256 to 4000 in 2d, and 800 to 3200 in 3d. 

In our previous report, several questions were said to be under investigation.  We have now answered all of those questions which concerned pinning susceptibility defined as:

In this equation,  p(φ, n, N) is the probability of a system jamming, with volume fraction φ, N particles and a fraction n =  N_f / N of fixed, pinned particles. In the limit of arbitrarily large N,  χ_p is expected to diverge - in analogy with the susceptiblity of a ferromagnetic or percolation system - as the distance to the jamming threshold,  Δφ , approaches zero:

            Question 1:  Does critical behavior hold in higher dimensions, and is the exponent γ_p = 1.2 universal?   Figures 2a and 2b show that in both 2d and 3d, susceptibility becomes more and more peaked around the (N-dependent) critical density φ_j .  Figures 3a and 3b show that the scaling form for the susceptibility in both 2d and 3d is consistent with this universal value.  This also supports the notion that 2d is the so-called "upper critical dimension" for jamming.

            Question 2: Is the finite-size scaling of the susceptibility best represented in terms of a diverging length scale like box width, L, or (as with other systems above their upper critical dimension) as fundamentally dependent on N ?  Thanks to the 3d simulations, we see that the latter is true, as Figures 3a and 3b show. In both cases, x = 0.5 .

            Question 3: Does isostaticity hold in the presence of pins; so that at the jamming threshold, the number of bonds present is precisely equal to the number of unconstrained degrees of freedom for the mobile particles? This answer is also "yes", with the caveat that even one fixed particle (whose position is frozen before equilibration) will break the translational invariance of the system. One expects that the number of excess bonds is:

where y = d if there is translational invariance, and y = 0 otherwise. This equation is supported in all cases by our data, with Figure 4 the case of N=600 in 2d.  Data were binned according to pressure, P, with zero pressure representing the jamming threshold.

            A final result concerns a two-variable scaling ansatz.  The notion that pinning disorder creates a new axis of the jamming phase diagram[2] suggests that there is an additional, independent way to approach the critical point.  Thus a universal scaling surface exists, which can be traced out by approaching along either the axis of rescaled volume fraction, or a combination of volume fraction and disorder. Figure 5 shows a view of this surface

            In future, we hope to pursue other questions raised by results from our first two years of this grant. For example: Which regular lattice geometries best support jamming in 2d and 3d, and why?  Will some key feature of Point J be altered when obstacles exceed some critical density of their own?  Other issues, related to the flow of both physical and self-propelled particles through apertures, were raised in our grant application but have not yet been tackled. It is hoped that we can create time to write codes to address important problems of this type.

            During the second fiscal year of this grant (September, 2013 - August, 2014) the remaining undergraduate who began work in the first grant year continued during the academic semester - unusual in our department, and a testament to his interest in the project. He presented a poster at our annual Sigma Xi sponsored poster session, and was a coauthor on a talk given at the American Physical Society (APS) meeting in March 2014.  A second coauthor is a former undergraduate student on our project, now a PhD student (Cornell University).

            The ACS PRF grant continues to help me, the PI, via continued conversation and collaboration with experts at the University of Pennsylvania and elsewhere.  I both contributed a talk at the APS March meeting 2014 and was asked to Chair the session. In the recent grant year, I gave my first invited talk on this research; it is hoped that further invitations will follow.

References:

[1] "Solids between the mechanical extremes of order and disorder", C.P. Goodrich, A.J. Liu and S.R. Nagel, Nature Physics 10, 578-581 (2014).   

[2] "Jamming in systems with quenched disorder", C.J. Olson Reichhardt, E. Groopman, Z. Nussinov, and C. Reichhardt,
 Phys. Rev. E  86, 061301 (2012).