In chapter 2, which is published in [117], we show that the transverse length scale in jammed packings can be understood, in analogy with `∗, as a length scale which
controls the stability of these packings to perturbations of their boundary. Unlike
`∗ which dictates whether or not a jammed system will be stable with respect to the replacement of periodic boundary conditions by free boundary conditions, the role of `T is more subtle. We show that `T determines whether or not a jammed system will be stable if particles on its boundary are allowed to deform in a manner that does not respect the system’s periodic boundary conditions.
Chapter 3, which is published in [30], discusses to what extent correlations of fluctuations in particle positions may be used to reliably reconstruct the dynamical matrix in experiment. To do this we consider a hexagonal colloidal crystal and find that the density of states identified in this way lacks the Van-Hove singularities characteristic of crystalline systems. From theory and comparison to simulation we show that the eigenvalues of the dynamical matrix tend to converge slowly as N/T where N is the number of degrees of freedom and T is the number of independent frames used in the analysis. This shows that great care must be taken when attempting to reconstruct the dynamical matrix from particle fluctuations to ensure that enough independent frames are acquired. Finally, we show that in a defected colloidal polycrystal the low frequency vibrational modes can be used to identify defects in an analogous manner to soft spots in amorphous solids. This work led to a second paper [127] that used the speeds of sound measured in colloidal experiments to deduce the elastic constants in these systems.
We then turn to chapter 4, which is published in [129], where we consider a well known method of measuring elastic moduli from correlations of particle fluctuations that was first introduced by Schall et al. [113]. Here a local strain tensor, Λαβ is constructed from particle fluctuations in a small volume. It is assumed that fluctuations of Λαβ have their origin in an elastic free energy of the formP(Λαβ)∼ exp(−µΛ2αβ/kBT) where µ is the elastic constant. While this method has been proven rigorously to yield accurate elastic constants in crystals we show that it fails when applied to disordered systems.
In chapter 5, which may be found in [115], we consider low temperature amor- phous solids under shear and show that the method of identifying soft spots using low-frequency quasi-localized modes even at finite temperatures and strain rates as well as for potentials beyond the standard harmonic disks. Moreover by tracking individual soft spots in time we show that each soft spot has a well defined lifetime that may be computed directly from the simulation. By studying the distribution of single-soft-spot lifetimes we are able to describe the flow of the solid in a sim- ple mean field picture. This work is related to an additional publication [109] in which we showed that the low-frequency vibrational modes could be applied to sin- gle defects in crystals, grain boundaries in polycrystals, and are also related to the direction of displacement experienced by particles during a rearrangement.
Chapters 6 and 7 both focus on the application of machine learning methods to identify soft spots. We begin by discussing the methodology, in chapter 6, and then applying it to the same sheared Lennard-Jones glass as in chapter 5 as well as a three-dimensional quiescent glass and a two dimensional experimental granular pil- lar under uniaxial compression. We show that this method yields a more accurate population of defects than those constructed with vibrational modes. Furthermore, the success of our method on a granular system shows that machine learning may be used to find defects even in systems that are not governed by a hamiltonian (and hence where vibrational modes are fundamentally inapplicable.) In chapter 7 we extend our method from finding soft spots as a binary classification to constructing a continuous softness field. We use this softness field to find a simple explanation
for dynamical heterogeneities in supercooled liquids based on a heterogeneous dis- tribution of energy barriers to rearrangement in these systems. Finally, we use a simple model to show that the relaxation of the glass as a whole can be recovered from the dynamics of the softness field. These results are published in [43, 116].
Chapter 2
Linear stability to continuous
boundary deformations
2.1
Introduction
At the jamming transition of ideal spheres, the removal of a single contact causes the rigid system to become mechanically unstable. [65, 91, 97] Thus at the transition, the replacement of periodic-boundary conditions with free-boundary conditions de- stroys rigidity even in the thermodynamic limit. [66, 150, 151] Recognizing that packings at the jamming threshold are susceptible to boundary conditions, Torquato and Stillinger [136] drew a distinction between collectively jammed packings, which are stable when the confining box is not allowed to deform, and strictly jammed
packings, which are stable to arbitrary perturbations of the boundary. Indeed, Dagois-Bohy et al. [46] have shown that jammed packings with periodic-boundary conditions can be linearly unstable to shearing the box.
At densities greater than the jamming transition there are more contacts than the minimum required for stability. [91, 97] In this regime one would expect pack- ings that are sufficiently large to be stable to changes in the boundary. How does the characteristic size for a stable system depend on proximity to the jamming tran- sition? The sensitivity to free- versus periodic-boundary conditions is governed by a length scale, `∗, that diverges at jamming transition. For L `∗, the system is
stable even with free boundaries. [66, 150, 151]
Our aim in this chapter is to identify the range of system pressures and sizes over which the system is likely to be unstable to a more general class of boundary perturbations in which particle displacements violate periodic boundary conditions. We will show that stability is governed by a competition between transverse plane waves and the anomalous modes that are unique to disordered systems. Thus, we show that stability for a large class of boundary perturbations is governed by a length scale,`T, that also diverges at the jamming transition. Packings withL`T
are linearly stable with respect to these boundary perturbations. We understand this as a competition between jamming transition physics at low pressures/system sizes, and transverse acoustic wave physics at high pressures/system sizes. The two lengths, `∗ and `
associated with the normal modes of jammed sphere packings. [123] We will discuss the physical meaning of these length scales in more detail in sec. 2.7.
We analyze athermal, frictionless packings with periodic boundary conditions composed of equal numbers of small and large spheres with diameter ratio 1:1.4 all with equal mass, m. The particles interact via the repulsive finite-range harmonic pair potential V(rij) = ε 2 1− rij σij 2 (2.1.1)
if rij < σij and V(rij) = 0 otherwise. Here rij is the distance between particles i and j, σij is the sum of the particles’ radii, and ε determines the strength of the potential. Energies are measured in units of ε, distances in units of the average particle diameter σ, and frequencies in units of pε/mσ2. We varied the total
number of particles from N = 32 to N = 512 at 36 pressures between p = 10−1
and p = 10−8. Particles are initially placed at random in an infinite temperature,
T =∞, configuration and are then quenched to aT = 0 inherent structure using a combination of linesearch methods, Newton’s method and the FIRE algorithm.[18] The resulting packing is then compressed or expanded uniformly in small increments until a target pressure,p, is attained. After each increment ofp, the system is again quenched to T = 0.