Date of Award

August 2020

Degree Type


Degree Name

Doctor of Philosophy (PhD)




Jennifer M. Schwarz


chromatin dynamics, jamming transition, pebble game, rigidity percolation

Subject Categories

Physical Sciences and Mathematics


indicated by the title, this is a thesis in two acts. The first act encapsulates a study of the rigidity transition in frictional particle packings, or how a packing goes from floppy-to-rigid as its particle density increases, for example. While consensus has emerged regarding the nature of the rigidity transition in frictionless packings, there is much less consensus with frictional packings. Therefore, I introduce two new complementary concepts, frictional rigidity percolation and minimal rigidity proliferation, to help identify the nature of the frictional rigidity transition. To probe frictional rigidity percolation, I construct rigid clusters using a (3,3) pebble game for sliding and frictional contacts first on a honeycomb lattice with next-nearest neighbors, and second on a hierarchical lattice. For both lattices, I find a continuous rigidity transition. My numerically obtained transition exponents for frictional rigidity percolation on the honeycomb lattice are distinct from those of frictionless/central-force rigidity percolation. I propose that localized motifs, such as hinges connecting rigid clusters that are allowed only with friction, could give rise to this new frictional universality class. I also develop a minimally rigid cluster generating algorithm invoking generalized Henneberg moves, dubbed minimal rigidity proliferation. For both frictional and central-force rigidity percolation, these clusters appear to be in the same universality class as connectivity percolation, suggesting superuniversality between all three transitions for such minimally rigid clusters. These combined results allow me to directly compare two universality classes on the same lattice in rigidity percolation, for the first time.

Grounded in this lattice work, I then turn towards identifying and analyzing rigid clusters within experimental packings to determine what aspects of the simpler lattice models are experimentally relevant. I use two approaches to identify the rigid clusters. Both approaches, the force-based dynamical matrix and the coordination-based rigidity percolation, agree with each other and identify similar rigid structures. As the system becomes jammed, at a contact number of $z=2.4\pm0.1$, a rigid backbone interspersed with floppy, particle-filled holes of a broad range of sizes emerges, creating a sponge-like morphology. I also find that the pressure within rigid structures always exceeds the pressure outside the rigid structures, i.e. that the backbone is load-bearing. These findings show that continuous transition observed in the lattice models persists in experiments and that mechanical stability arises through arch structures and hinges at the mesoscale.

In the second act of this thesis, I turn towards biology for inspiration, namely biology in the form of the cell nucleus. The cell nucleus houses chromatin, which is linked to a protein shell called the lamina. Protein motors and chromatin binding proteins in the nucleus are thought to drive correlated chromatin dynamics and nuclear shape fluctuations. To test this notion, we develop a minimalistic model in which an active, crosslinked Rouse chain linked to a polymeric shell. System-scale correlated motion occurs and requires both motor activity and crosslinks. Contractile motors, in particular, enhance chromatin dynamics by driving anomalous density fluctuations. Nuclear shape fluctuations depend on motor strength, crosslinking, and chromatin-lamina linkage. Complex chromatin dynamics and nuclear shape, therefore, both emerge from this minimal, yet composite, system.


Open Access