Force Distributions near Jamming and Glass Transitions

Abstract

We calculate the distribution of interparticle normal forces P (F ) near the glass and jamming transitions in model supercooled liquids and foams, respectively. P (F ) develops a peak that appears near the glass or jamming transitions, whose height increases with decreasing temperature, decreasing shear stress and increasing packing density. A similar shape of P (F ) was observed in experiments on static granular packings. We propose that the appearance of this peak signals the development of a yield stress. The sensitivity of the peak to temperature, shear stress and density lends credence to the recently proposed generalized jamming phase diagram. 64.70.Pf,81.05.Rm,83.70.Hq Granular materials can flow when shaken, but jam when the shaking intensity is lowered [1]. Similarly, foams and emulsions can flow when sheared, but jam when shear stress is lowered [2]. These systems are athermal because thermal energy is insufficient to change the packing of grains, bubbles, or droplets. When the external driving force is too small to cause particle rearrangements, these materials become amorphous solids and develop a yield stress. A supercooled liquid, on the other hand, is a thermal system that turns, as temperature is lowered, into a glass-an amorphous solid with a yield stress [3]. Despite significant differences between driven, athermal systems and quiescent, thermal ones, it has been suggested that the process of jamming-developing a yield stress in an amorphous state-may lead to common behavior, and that these systems can be unified by a jamming phase diagram [4]. This implies that there should be similarities in these different systems as they approach jamming or glass transitions. We test this speculation by measuring the distribution P (F ) of interparticle normal forces F , in model supercooled liquids and foams. We find that for glasses, P (F ) is quantitatively similar to experimental results on granular materials [5].When granular materials jam, the distribution of stresses is known to be inhomogeneous [6,7]. As proposed in Ref.[7], we quantify this effect by measuring P (F ). Our aim is to determine which feature in P (F ) is associated with development of a yield stress. Experiments [5,8] and simulations [9,10] on static granular packings find that P (F ) has a plateau or small peak at small F and decays exponentially at large F . We argue that the development of a peak is the signature of jamming.For supercooled liquids, equilibrium statistical mechanics gives insight into the shape of P (F ). Since forces depend only on particle separations, P (F )dF = G(r)dr, where G(r)dr is the probability of finding a particle between r and r + dr given a particle at the origin. Thus,, where N is the number of particles, ρ is the number density, g(r) is the pair distribution function, and S D r D−1 is the surface area of a Ddimensional sphere of radius r. Although it is well known that g(r) does not change significantly as the temperature is varied through the glass transition T g , we show below that P (F ) is qu...

Dynamical Heterogeneity and Jamming in Glass-Forming Liquids

“…An explanation of the origin of the long exponential tail in P (F ) was proposed in Ref. [1] and is related to the fact that large F behavior can be obtained from the small r behavior of g(r) ≈ y(r)exp[−V(r)/k b T], and P (F )dF ∝ g(r)dr. The exponential tail in the force distribution of granular materials was suggested in the q-model in Ref.…”

“…We define two LJ particles to be in contact if they interact with a positive (repulsive) force, as it was done in Ref. [1]. Commonly, granular materials are modeled as systems of particles with only repulsive interactions when grains are in contact.…”

“…We include P (F/ F c ) for several temperatures below T 0 = 0.48 for comparison with the force distribution found in Ref. [1] below T g . As in Figure 2, the probability of observing forces substantially larger than F decays exponentially, and the curvature around F decreases as T decreases (See Figure 3 caption).…”

“…Several speculations have been made in Refs. [1,71] in order to explain the jamming scenario and its possible connection to the glass transition. They theorize that "systems jam when there are enough particles in a force chain network to support stress over the time scale of the measurement" which would imply that force chains observed in granular packing may also be important to the glass transition.…”

“…Following Ref. [1], we calculate the average force F between every pair of neighboring particles at a given instant of time. In Figure 1 we show F vs T calculated from at least N conf = 1000 independent N V E configurations at each T .…”