Yielding in an Integer Automaton Model for Amorphous Solids under Cyclic Shear.

We present results on an automaton model of an amorphous solid under cyclic shear. After a transient, the steady state falls into one of three cases in order of increasing strain amplitude: (i) pure elastic behavior with no plastic activity, (ii) limit cycles where the state recurs after an integer period of strain cycles, and (iii) irreversible plasticity with longtime diffusion. The number of cycles N required for the system to reach a periodic orbit diverges as the amplitude approaches the yielding transition between regimes (ii) and (iii) from below, while the effective diffusivity D of the plastic strain field vanishes on approach from above. Both of these divergences can be described by a power law. We further show that the average period T of the limit cycles increases on approach to yielding.

Avalanches, thresholds, and diffusion in mesoscale amorphous plasticity.

We present results on a mesoscale model for amorphous matter in athermal, quasistatic (a-AQS), steady-state shear flow. In particular, we perform a careful analysis of the scaling with the lateral system size L of (i) statistics of individual relaxation events in terms of stress relaxation S, and individual event mean-squared displacement M, and the subsequent load increments Δγ, required to initiate the next event; (ii) static properties of the system encoded by x=σ_{y}-σ, the distance of local stress values from threshold; and (iii) long-time correlations and the emergence of diffusive behavior. For the event statistics, we find that the distribution of S is similar to, but distinct from, the distribution of M. We find a strong correlation between S and M for any particular event, with S∼M^{q} with q≈0.65. The exponent q completely determines the scaling exponents for P(M) given those for P(S). For the distribution of local thresholds, we find P(x) is analytic at x=0, and has a value P(x)|_{x=0}=p_{0} which scales with lateral system length as p_{0}∝L^{-0.6}. The size dependence of the average load increment 〈Δγ〉 appears to be asymptotically controlled by the plateau behavior of P(x) rather than by a subsequent apparent power-law behavior. Extreme value statistics arguments lead thus to a scaling relation between the exponents governing P(x) and those governing P(S). Finally, we study the long-time correlations via single-particle tracer statistics. The value of the diffusion coefficient is completely determined by 〈Δγ〉 and the scaling properties of P(M) (in particular from 〈M〉) rather than directly from P(S) as one might have naively guessed. Our results (i) further define the a-AQS universality class, (ii) clarify the relation between avalanches of stress relaxation and diffusive behavior, and (iii) clarify the relation between local threshold distributions and event statistics.

Diffusion in Mesoscopic Lattice Models of Amorphous Plasticity.

We present results on tagged particle diffusion in a mesoscale lattice model for sheared amorphous material in athermal quasistatic conditions. We find a short time diffusive regime and a long time diffusive regime whose diffusion coefficients depend on system size in dramatically different ways. At short time, we find that the diffusion coefficient, D, scales roughly linearly with system length, D∼L^{1.05}. This short time behavior is consistent with particle-based simulations. The long-time diffusion coefficient scales like D∼L^{1.6}, close to previous studies which found D∼L^{1.5}. Furthermore, we show that the near-field details of the interaction kernel do not affect the short time behavior but qualitatively and dramatically affect the long time behavior, potentially causing a saturation of the mean-squared displacement at long times. Our finding of a D∼L^{1.05} short time scaling resolves a long standing puzzle about the disagreement between the diffusion coefficient measured in particle-based models and mesoscale lattice models of amorphous plasticity.

Paramagnetic colloids: Chaotic routes to clusters and molecules.

We present computer simulations and experiments on dilute suspensions of superparamagnetic particles subject to rotating magnetic fields. We focus on chains of four particles and their decay routes to stable structures. At low rates, the chains track the external field. At intermediate rates, the chains break up but perform a periodic (albeit complex) motion. At sufficiently high rates, the chains generally undergo chaotic motion at short times and decay to either closely packed clusters or more dispersed, colloidal molecules at long times. We show that the transition out of the chaotic states can be described as a Poisson process in both simulation and experiment.

Assembling particle clusters with incoherent 3D magnetic fields.

Directed assembly of particle suspensions in massively parallel formats, such as with magnetic fields, has application in rheological control, smart drug delivery, and active colloidal devices from optical materials to microfluidics. At the heart of these applications lies a control optimization problem for driving the assembly and dissolution of highly monodisperse particle clusters. For magnetic field control, most attention to-date has been centered around in-phase coherent magnetic fields. Instead, we investigate a family of incoherent 3D magnetic fields that are capable of creating controlled and tunable particle assemblies such as dimers, trimers, and quadramers. These field functions can be tuned to assemble monodisperse clusters with long term stability and can quickly switch the clusters between different states. This subset of three-dimensional field functions that we have studied demonstrates the rich phase space available to tune colloidal suspensions with magnetic fields.

Anomalous stress fluctuations in athermal two-dimensional amorphous solids.

We numerically study the local stress distribution within athermal, isotropically stressed, mechanically stable, packings of bidisperse frictionless disks above the jamming transition in two dimensions. Considering the Fourier transform of the local stress, we find evidence for algebraically increasing fluctuations in both isotropic and anisotropic components of the stress tensor at small wave numbers, contrary to recent theoretical predictions. Such increasing fluctuations imply a lack of self-averaging of the stress on large length scales. The crossover to these increasing fluctuations defines a length scale ℓ_{0}, however, it appears that ℓ_{0} does not vary much with packing fraction ϕ, nor does ℓ_{0} seem to be diverging as ϕ approaches the jamming ϕ_{J}. We also find similar large length scale fluctuations of stress in the inherent states of a quenched Lennard-Jones liquid, leading us to speculate that such fluctuations may be a general property of amorphous solids in two dimensions.

Gelation and mechanical response of patchy rods.

We perform Brownian dynamics simulations to study the gelation of suspensions of attractive, rod-like particles. We show that in detail the rod-rod surface interactions can dramatically affect the dynamics of gelation and the structure and mechanics of the networks that form. If the attraction between the rods is perfectly smooth along their length, they will collapse into compact bundles. If the attraction is sufficiently corrugated or patchy, over time, a rigid space-spanning network will form. We study the structure and mechanical properties of the networks that form as a function of the fraction of the surface, f, that is allowed to bind. Surprisingly, the structural and mechanical properties are non-monotonic in f. At low f, there are not a sufficient number of cross-linking sites to form networks. At high f, rods bundle and form disconnected clusters. At intermediate f, robust networks form. The elastic modulus and yield stress are both non-monotonic in the surface coverage. The stiffest and strongest networks show an essentially homogeneous deformation under strain with rods re-orienting along the extensional axis. Weaker, more clumpy networks at high f re-orient relatively little with strong non-affine deformation. These results suggest design strategies for tailoring surface interactions between rods to yield rigid networks with optimal mechanical properties.

Elasticity of frictionless particles near jamming.

We study the linear elastic response of harmonic disk packings near jamming via three types of probes: (i) point forcing, (ii) constrained homogeneous deformation of subregions of large systems, and (iii) unconstrained deformation of the full system subject to periodic boundary conditions. For the point forcing, our results indicate that the transverse component of the response is governed by a lengthscale ξT, which scales with the confining pressure, p, as ξT∼p-0.25, while the longitudinal component is governed by ξL, which scales as ξL∼p-0.4. The former scaling is precisely the transverse lengthscale, which has been invoked to explain the structure of normal modes near the density of states anomaly in sphere packings, while the latter is much closer to the rigidity length, l*∼p-0.5, which has been invoked to describe the jamming scenario. For the case of constrained homogeneous deformation, we find that µ(R), the value of the shear modulus measured in boxes of size R, gives a value much higher than the continuum result for small boxes and recedes to its continuum limit only for boxes bigger than a characteristic length, which scales like p-0.5, precisely the same way as l*. Finally, for the case of unconstrained homogeneous deformation, we find displacement fields with power spectra, which are consistent with independent, uncorrelated Eshelby transformations. The transverse sector is amazingly invariant with respect to p and very similar to what is seen in Lennard-Jones glasses. The longitudinal piece, however, is sensitive to p. It develops a plateau at long wavelength, the start of which occurs at a length that grows in the p→0 limit. Strikingly, the same behavior is observed both for applied shear and dilation.

Toward Accumulation of Magnetic Nanoparticles into Tissues of Small Porosity.

Magnetic concentration of drug-laden magnetic nanoparticles has been proven to increase the delivery efficiency of treatment by 2-fold. In these techniques, particles are concentrated by the presence of a magnetic source that delivers a very high magnetic field and a strong magnetic field gradient. We have found that such magnetic conditions cause even 150 nm particles to aggregate significantly into assemblies that exceed several micrometers in length within minutes. Such assembly sizes exceed the effective intercellular pore size of tumor tissues preventing these drug-laden magnetic nanoparticles from reaching their target sites. We demonstrate that by using dynamic magnetic fields instead, we can break up these magnetic nanoparticles while simultaneously concentrating them at target sites. The dynamic fields we investigate involve precessing the field direction while maintaining a field gradient. Manipulating the field direction drives the particles into attractive and repulsive configurations that can be tuned to assemble or disassemble these particle clusters. Here, we develop a simple analytic model to describe the kinetic thresholds of disassembly and we compare both experimental and numerical results of magnetic particle suspensions subjected to dynamic fields. Finally we apply these methods to demonstrate penetration in a porous scaffold with a similar pore size to that expected of a tumor tissue.

Avalanches in strained amorphous solids: does inertia destroy critical behavior?

Simulations are used to determine the effect of inertia on athermal shear of amorphous two-dimensional solids. In the quasistatic limit, shear occurs through a series of rapid avalanches. The distribution of avalanches is analyzed using finite-size scaling with thousands to millions of disks. Inertia takes the system to a new underdamped universality class rather than driving the system away from criticality as previously thought. Scaling exponents are determined for the underdamped and overdamped limits and a critical damping that separates the two regimes. Systems are in the overdamped universality class even when most vibrational modes are underdamped.

Energy barrier scalings in driven systems.

Energy landscape mappings are performed for two different molecular systems under mechanical loads. Barrier heights are observed to scale as Delta U approximately delta(3/2), where delta is a residual load. Catastrophe theory predicts that this scaling should arise for vanishing delta; however, this region is irrelevant in physical processes at finite temperature because thermal fluctuations cause the system to cross over the barrier before reaching the small-delta regime. Surprisingly, we find that the Delta U approximately delta(3/2), scaling is valid far beyond the vanishing delta regime described by catastrophe theory. We discuss how this scaling will therefore be relevant at finite temperatures and gives corrections to Eyring's theory for transition rates.

Amorphous systems in athermal, quasistatic shear.

We present results on a series of two-dimensional atomistic computer simulations of amorphous systems subjected to simple shear in the athermal, quasistatic limit. The athermal quasistatic trajectories are shown to separate into smooth, reversible elastic branches which are intermittently broken by discrete catastrophic plastic events. The onset of a typical plastic event is studied with precision, and it is shown that the mode of the system which is responsible for the loss of stability has structure in real space which is consistent with a quadrupolar source acting on an elastic matrix. The plastic events themselves are shown to be composed of localized shear transformations which organize into lines of slip which span the length of the simulation cell, and a mechanism for the organization is discussed. Although within a single event there are strong spatial correlations in the deformation, we find little correlation from one event to the next, and these transient lines of slip are not to be confounded with the persistent regions of localized shear--so-called "shear bands"--found in related studies. The slip lines give rise to particular scalings with system length of various measures of event size. Strikingly, data obtained using three differing interaction potentials can be brought into quantitative agreement after a simple rescaling, emphasizing the insensitivity of the emergent plastic behavior in these disordered systems to the precise details of the underlying interactions. The results should be relevant to understanding plastic deformation in systems such as metallic glasses well below their glass temperature, soft glassy systems (such as dense emulsions), or compressed granular materials.