Jamming in soft disk packings

This thesis considers the mechanical properties of amorphous solids such as foams, emulsions, and granular media. Each of these systems consists of “particles” (bubbles, droplets, grains) in a dense, disordered structure. As a system is compressed, it eventually forms enough contacts between particles that it can support a load, including shearing stresses. We say that it has jammed. A major theoretical challenge is to describe material properties in the vicinity of this non-equilibrium phase transition. Jamming has been widely studied in the context of a specific model, namely non- Brownian packings of soft frictionless disks or spheres. The particles repel when they overlap, and otherwise do not interact. They come in two distinct sizes to prevent crystallization; by convention the concentration and diameter ratio of the species are fixed to specific values. Little is known about the nature of the jammed solids that result when these restrictions are lifted. This is a significant knowledge gap, because bubbles, droplets, and grains routinely experience some degree of attraction to their neighbors (e.g. due to depletion interactions or capillary bridges), and their size distribution can vary considerably. Hence the goal of this thesis is to determine how the soft sphere model jams (i) when the degree of attraction between particles is varied, and (ii) when the size and number ratio of particles in a repulsive bidisperse packing is varied. Attraction.—First, we study how soft particles with an attractive shell become rigid. By analyzing the percolation of rigid clusters of particles, we present evidence for two distinct jamming scenarios. Strongly attractive systems undergo a continuous transition in which rigid clusters grow and ultimately diverge in size at a critical packing fraction. Purely repulsive and weakly attractive systems jam via a first order transition, with no growing cluster size. We further show that the weakly attractive scenario is a finite size effect, so that for any nonzero attraction strength, a sufficiently large system will fall in the strongly attractive universality class. We therefore expect attractive jamming to be generic in the laboratory and in nature. Second, we probe the elasticity of the strongly attractive solid. By treating the jamming point as a critical point, we exploit critical scaling analysis to determine the shear modulus, bulk modulus, and coordination of marginal solids close to the sticky jamming point. We find that each observable differs not just quantitatively but also qualitatively from the purely repulsive case. Size and number ratio.—We systematically map out the jamming transition of 2D bidisperse mixtures of disks in the hard particle limit. The critical volume fraction and multiple structural and mechanical properties all show a rich variation with mixture composition and particle size ratio, and can therefore be tuned by choosing certain mixtures. We identify two local minima in the critical volume fraction, both of which have low structural order; one minimum is close to the widely studied 50:50 mixture of particles with a ratio of radii of 1:1.4. We also identify a region at low size ratios characterized by increased structural order, with a corresponding enhancement in the stiffness.