TY - JOUR
T1 - Force distribution affects vibrational properties in hard-sphere glasses
JF - Proceedings of the National Academy of Sciences
JO - Proc Natl Acad Sci USA
SP - 17054
LP - 17059
DO - 10.1073/pnas.1415298111
VL - 111
IS - 48
AU - DeGiuli, Eric
AU - Lerner, Edan
AU - Brito, Carolina
AU - Wyart, Matthieu
Y1 - 2014/12/02
UR - http://www.pnas.org/content/111/48/17054.abstract
N2 - How a liquid becomes rigid at the glass transition is a central problem in condensed matter physics. In many scenarios of the glass transition, liquids go through a critical temperature below which minima of free energy appear. However, even in the simplest glass, hard spheres, what confers mechanical stability at large density is highly debated. In this work we show that to quantitatively understand stability at a microscopic level, the presence of weakly interacting pairs of particles must be included. This approach allows us to predict various nontrivial scaling behavior of the elasticity and vibrational properties of colloidal glasses that can be tested experimentally. It also gives a spatial interpretation to recent, exact calculations in infinite dimensions.We theoretically and numerically study the elastic properties of hard-sphere glasses and provide a real-space description of their mechanical stability. In contrast to repulsive particles at zero temperature, we argue that the presence of certain pairs of particles interacting with a small force f soften elastic properties. This softening affects the exponents characterizing elasticity at high pressure, leading to experimentally testable predictions. Denoting ℙ(f)∼fθe, the force distribution of such pairs and ϕc the packing fraction at which pressure diverges, we predict that (i) the density of states has a low-frequency peak at a scale ω*, rising up to it as D(ω)∼ω2+a, and decaying above ω* as D(ω)∼ω−a where a=(1−θe)/(3+θe) and ω is the frequency, (ii) shear modulus and mean-squared displacement are inversely proportional with 〈δR2〉∼1/μ∼(ϕc−ϕ)κ, where κ=2−2/(3+θe), and (iii) continuum elasticity breaks down on a scale ℓc∼1/δz∼(ϕc−ϕ)−b, where b=(1+θe)/(6+2θe) and δz=z−2d, where z is the coordination and d the spatial dimension. We numerically test (i) and provide data supporting that θe≈0.41 in our bidisperse system, independently of system preparation in two and three dimensions, leading to κ≈1.41, a≈0.17, and b≈0.21. Our results for the mean-square displacement are consistent with a recent exact replica computation for d=∞, whereas some observations differ, as rationalized by the present approach.
ER -