Browsing by Author "Socolar, Joshua"

What affects the transition of a collection of grains from flowing to a rigid packing? Previous efforts towards answering this important question have led to various versions of ``jamming’’ phase diagrams, which specify conditions under which a granular material behaves like solid, i.e., in a jammed phase. In this dissertation, we report two sets of experiments to study the influence of particle shape and of the form of the applied shear strain on the jamming phase diagram of slowly deformed frictional granular materials. We use 2d photoelastic particles to measure the overall pressure of the system and various physical quantities that characterize the contact network such as the averaged number of contacts per particle.

In the first set of experiments, we systematically compare the mechanical and geometrical properties of uniaxially compressed granular materials consisting of particles with shapes of either regular pentagon or disk. The compression is applied quasi-statically and induces a density-driven jamming transition. We find that pentagons and disks jam at similar packing fraction. At the onset of jamming, disks have contact numbers consistent with predictions from an ideal constraint counting argument. However, this argument fails to predict the right contact number for pentagons. We also find that both jammed pentagons and disks show the Gamma distribution of the Voronoi cell area with the same parameters. Moreover, jammed pentagons have similar translational order for particle centers but slightly less orientational order for contacting pairs compared to jammed disks. Finally, we report observations that for jammed pentagons, the angle between edges at a face-to-vertex contact point shows a uniform distribution and the size of a cluster connected by face-to-face contacts shows a power-law distribution.

In the second set of experiments, we use a novel multi-ring Couette shear apparatus that we developed to eliminate shear banding which unavoidably appears in conventional Couette shear experiments. A shear band is a narrow region where a lot of rearrangements of particles occur. The shear band usually has a much smaller packing fraction than the rest of the system. We map out a jamming phase diagram experimentally, and for the first time perform a systematic direct test of the mechanical responses of the jammed states created by shearing under reverse shear. We find a clear distinction between fragile states and shear-jammed states: the latter do not collapse under reverse shear. The yield stress curve is also mapped out, which marks the stress needed for the shear-jammed states to enter a steady regime where many plastic rearrangements of particles happen and the overall stress fluctuates around a constant. Interestingly, for large packing fraction, a shear band still develops when the system remains strongly jammed in the steady regime. We find that the cooperative motion of particles in this regime is highly heterogeneous and can be quantified by a dynamical susceptibility, which keeps growing as the packing fraction increases.

Our observations not only serve as important data to construct theories to explain the origin of rigidity in density-driven jamming and shear-induced jamming but also are relevant to many other key problems in the physics of granular matter from the stability of a jammed packing to the complex dynamics of dense granular flows.

This dissertation presents three studies on Boolean networks. Boolean networks are a class of mathematical systems consisting of interacting elements with binary state variables. Each element is a node with a Boolean logic gate, and the presence of interactions between any two nodes is represented by directed links. Boolean networks that implement the logic structures of real systems are studied as coarse-grained models of the real systems. Large random Boolean networks are studied with mean field approximations and used to provide a baseline of possible behaviors of large real systems. This dissertation presents one study of the former type, concerning the stable oscillation of a yeast cell-cycle oscillator, and two studies of the latter type, respectively concerning the statistical complexity of large random Boolean networks and an extension of traditional mean field techniques that accounts for the presence of short loops.

In the cell-cycle oscillator study, a novel autonomous update scheme is introduced to study the stability of oscillations in small networks. A motif that corrects pulse-growing perturbations and a motif that grows pulses are identified. A combination of the two motifs is capable of sustaining stable oscillations. Examining a Boolean model of the yeast cell-cycle oscillator using an autonomous update scheme yields evidence that it is endowed with such a combination.

Random Boolean networks are classified as ordered, critical or disordered based on their response to small perturbations. In the second study, random Boolean networks are taken as prototypical cases for the evaluation of two measures of complexity based on a criterion for optimal statistical prediction. One measure, defined for homogeneous systems, does not distinguish between the static spatial inhomogeneity in the ordered phase and the dynamical inhomogeneity in the disordered phase. A modification in which complexities of individual nodes are calculated yields vanishing complexity values for networks in the ordered and critical phases and for highly disordered networks, peaking somewhere in the disordered phase. Individual nodes with high complexity have, on average, a larger influence on the system dynamics.

Lastly, a semi-annealed approximation that preserves the correlation between states at neighboring nodes is introduced to study a social game-inspired network model in which all links are bidirectional and all nodes have a self-input. The technique developed here is shown to yield accurate predictions of distribution of players' states, and accounts for some nontrivial collective behavior of game theoretic interest.

An important challenge in the science of granular materials is to understand the connection between the shapes of individual grains and the macroscopic response of the aggregate. Granular packings of concave or elongated particles can form free-standing structures like walls or arches, in sharp contrast to the behaviors of spherical grains. For some particle shapes, such as staples, the rigidity arises from interlocking of pairs of particles, but the origins of rigidity for non-interlocking particles remains unclear. In addition to their intrinsic interest, these packings are relevant to lightweight and reconfigurable structures in civil, geotechnical and material engineering applications.

In this thesis, we report on experiments and numerical simulations of packings of star-shaped particles consisting of three mutually orthogonal sphero-cylinders whose centers coincide. The first set of experiments studies the chance of obtaining a free-standing column when the confining tube of the column is removed, which we will call it as ``intrinsic stability''. We prepare monodisperse packings of star-shaped particles with different length-to-arm diameter aspect ratio α, interparticle friction and particle-base friction. We also vary packing density by vibrating the packings when they are in the tube. We find that the intrinsic stability depends on packing dimension: columns of greater diameter or shorter height are more stable. Both arm length and interparticle friction can greatly increase the intrinsic stability, while the packing density and basal friction have limited effects on the intrinsic stability.

The second set of experiments involves stability of free-standing columns (prepared from the first set of experiments) under three different external perturbations: (1) base tilting; (2) static axial loading; and (3) vertical vibration. For the base tilting test, we gradually tilt the base of the column and observe column collapse as a function of tilt angle. We find that columns of low friction particles are more fragile than those of high friction particles. For the axial loading test, we gradually increase the loading on a column until it collapses. We find that tall columns are more fragile. For the vibration test, we apply vertical sinusoidal vibration from the base to destabilize the column. Both interparticle and basal friction improve packing stability in terms of increasing relaxation time under vibration. We also find that tall columns are more sensitive to the vibration in the sense that they collapse faster than short ones under the same vibration.

In the third set of experiments, we vary α and subject the packings to quasistatic direct shear. For small α, we observe a finite yield stress. For large α, however, the packings become rigid when sheared, supporting stresses that increase sharply with increasing strain. Analysis of x-ray micro-computed tomography data collected during the shear reveals that the stiffening is associated with a tilted, oblate cluster of particles near the nominal shear plane in which particle deformation and average contact number both increase.

Molecular dynamics simulations that closely match the third experiments are used to investigate the finite yield stress and the stiffening. In simulation, interparticle contact forces are known to us. For yield packings (small α), simulations suggest no apparent cohesion. For stiffening packings (large α), simulation results show that the particles are collectively under tension along one direction even though they do not interlock pairwise. These tensions come from contact forces with large associated torques, and they are perpendicular to the compressive stresses in the packing. They counteract the tendency to dilate, thus stabilizing the particle cluster.

In a variety of physical systems, slow driving produces self-similar intermittent dynamics known as crackling noise. Barkhausen noise in ferromagnets, acoustic emission in fracture, seismic activities and failure in sheared granular media are few examples of crackling dynamics with substantial differences at the microscopic scale but similar universal laws. In many of the crackling systems, the origin of this universality and the connection between microscopic and macroscopic scales are subjects of current investigations.

We perform experiments to study the microscopic and macroscopic dynamics of a sheared granular medium. In our experiments, a constant speed stage pulls a slider with a loading spring across a 2D granular medium. We measure the pulling force on the spring, and image the medium to extract the local stress and particle displacement. Using novel signal and image analysis methods, we identify fast energy dissipating events, i.e.\ avalanches, and investigate their statistics and dynamics.

The pulling force exhibits crackling dynamics for low driving rates with intermittent slip avalanches. The energy loss in the spring has a power-law distribution with an exponent that strongly depends on the driving rate and is different from $-1.5$ predicted by several models. In our experiments for low driving rate, we find a slip rate power-spectrum of form $\mathcal{P}_v(\omega) \sim \frac{\omega^2}{1+\omega^{2.4}}$, a power-law distribution of the slip rate $P(v) \sim v^{-2.9}$, and average temporal profile of the slider motion (avalanche shape) of form $\mathcal{P}_D(u)=[u(1-u)]^{1.09}$. These findings are different from several theoretical and numerical studies \citep{dahmen2011simple, colaiori2008exactly, Laurson13_natcom}.

Avalanche temporal correlation is also investigated using certain conditional probabilities. At low driving rates, we observe uncorrelated order of the avalanches in terms of Omori-Utsu and B\r{a}th laws and temporal correlation in terms of the waiting time law. At higher driving rates, where the sequence of slip avalanches shows strong periodicity, we observe scaling laws and asymmetrical avalanche shapes that are clearly distinguishable from those in the crackling regime. We provide a novel dynamic phase diagram of granular matter as a function of driving rate and stiffness and characterize the crackling to periodic transition. We also find intermittent fluctuations in internal stress both in the crackling and the periodic regime.

Finally, we observe a narrow shear band with most of particle displacements, but stress fluctuations all over the medium. We identify the spatio-temporal connected components of local stress drops, which we call local avalanches. We find power-law distributions of the local avalanches with an exponent of $-1.7 \pm 0.1$, different from spring energy avalanche distribution with an exponent of $-0.41 \pm 0.05$ for the same experiments.

Our study constrains theoretical frameworks for granular dynamics and crackling noise in sheared granular media. Moreover, it may be relevant for characterizing the role of granular matter in fault gouges during seismic events.

One of the main challenges that biological oscillators face at the cellular level is maintaining coherence in the presence of molecular noise. Mechanisms of noise resistance have been proposed, however the findings are sometimes contradictory and not universal. Another challenge faced by biological oscillators is the proper timing of cellular events and effective distribution of cellular resources when there is more than one oscillator in the same cell. Biological oscillators are often coupled, however, the mechanisms and extent of these couplings are poorly understood. In this thesis, I describe three separate yet interconnected projects in an attempt to understand these biophysical phenomena.

I show that slow DNA unbinding rates are important in titration-based oscillators and can mitigate molecular noise. Multiple DNA binding sites can also increase the coherence of the oscillations through protected states, where the DNA binding/unbinding between these states has little effect on gene expression. I then show that experimental titration-based oscillator in budding yeast is innately coupled to the cell cycle. The oscillator and the cell cycle show 1:1 and 2:1 phase locking similar to what has been observed in natural systems. Finally, by studying the relationship between the circadian redox rhythm and genetic circadian clock in plants I show how perturbation of one of the coupled oscillators can be transformed into a reinforcement signal for the other one via a balanced network architecture.

During embryogenesis, a single zygote gives rise to a multicellular embryo with distinct spatial territories marked by differential gene expression. How is this patterning process organized? How robust is this function to perturbations? Experiments that examine normal and regulative development will provide direct evidence for reasoning out the answers to these fundamental questions. Recent advances in technology have led to experimental determinations of increasingly complex gene regulatory networks (GRNs) underlying embryonic development. These GRNs offer a window into systems level properties of the developmental process, but at the same time present the challenge of characterizing their behavior. A suitable modeling framework for developmental systems is needed to help gain insights into embryonic development. Such models should contain enough detail to capture features of interest to developmental biologists, while staying simple enough to be computationally tractable and amenable to conceptual analysis. Combining experiments with the complementary modeling framework, we can grasp a systems level understanding of the regulatory program not readily visible by focusing on individual genes or pathways.

This dissertation addresses both modeling and experimental challenges. First, we present the autonomous Boolean network modeling framework and show that it is a suitable approach for developmental regulatory systems. We show that important timing information associated with the regulatory interactions can be faithfully represented in autonomous Boolean models in which binary variables representing expression levels are updated in continuous time, and that such models can provide direct insight into features that are difficult to extract from ordinary differential equation (ODE) models. As an application, we model the experimentally well-studied network controlling fly body segmentation. The Boolean model successfully generates the patterns formed in normal and genetically perturbed fly embryos, permits the derivation of constraints on the time delay parameters, clarifies the logic associated with different ODE parameter sets, and provides a platform for studying connectivity and robustness in parameter space. By elucidating the role of regulatory time delays in pattern formation, the results suggest new types of experimental measurements in early embryonic development. We then use this framework to model the much more complicated sea urchin endomesoderm specification system and describe our recent progress on this long term effort.

Second, we present experimental results on developmental plasticity of the sea urchin embryo. The sea urchin embryo has the remarkable ability to replace surgically removed tissues by reprogramming the presumptive fate of remaining tissues, a process known as transfating, which in turn is a form of regulative development. We show that regulative development requires cellular competence, and that competence is lost early on but can be regained after further differentiation. We demonstrate that regulative replacement of missing tissues can induce distal germ layers to participate in reprogramming, leading to a complete re-patterning in the remainder of the embryo. To understand the molecular mechanism of cell fate reprogramming, we examined micromere depletion induced non-skeletogenic mesoderm (NSM) transfating. We found that the skeletogenic program was greatly temporally compressed in this case, and that akin to another NSM transfating case, the transfating cells went through a hybrid regulatory state where NSM and skeletogenic marker genes were co-expressed.

An important challenge in the physics of granular materials is understanding how the properties of single grains, such as grain shape and friction, influence the mechanical strength and dynamical response of the bulk granular material. While spherical grains are often used to study granular materials in experiments and simulations, the interactions among grains, and in many cases the flow and stability of granular packings, change when grain shape is modified. In this dissertation, we explore the influence of friction and grain shape on grain-scale dynamics, properties of mesoscale force chains, and macroscopic stick-slip dynamics of granular materials through novel experiments. In one set of experiments, an intruder is pushed by a spring through an annular cell filled with a quasi-2D monolayer of photoelastic grains that either contact a glass substrate or float on water. We characterize the effects of basal friction between the substrate and grains, intergrain friction, intruder size, and grain shape on the dynamics of the intruder, the flow of grains during slip events, and spatial distribution of stresses within the granular material in stable sticking periods. In another set of experiments, a slider is pulled by a spring across a quasi-2D monolayer of gravity-packed grains set between two glass plates. We observe the influence of grain angularity on statistical properties characterizing the stick-slip dynamics of the slider as well as grain-scale dynamics and stresses.

We first compare the dynamics of the intruder driven through packings of disks that either contact the glass base -- having basal friction -- or float on water -- having no basal friction. At high packing fractions, we find that the intruder exhibits stick-slip dynamics when basal friction acts on the grains, but the intruder instead flows freely through the granular material, with only occasional sticking periods (called intermittent flow or clogging-like dynamics, quantified by the average time between sticking periods), when basal friction is removed. We also observe when basal friction is present that the intruder's dynamics transition from stick-slip to intermittent flow with decreasing packing fraction; this transition occurs at a higher packing fraction with lower intergrain friction. Lastly, in simulations that model this experimental system, we vary static and dynamic basal friction coefficients and show that dynamic basal friction, rather than static basal friction, determines whether the intruder exhibits stick-slip or intermittent flow at high packing fractions.

We next vary the size of the intruder and, at several different packing fractions for each intruder size, compute statistics of the waiting time between sticking periods, duration of sticking periods, energy released in slip events, and force of grains acting on the intruder. We show that each statistical measure for all intruder sizes collapses to a single curve when packing fraction is rescaled by the packing fraction below which the intruder carves out a completely open channel in the granular material. With a geometrical model, we relate the packing fraction of open channel formation to a characteristic packing fraction of the material and the ratio of the intruder's diameter and the width of the annular cell, and we confirm the prediction of this model.

We thirdly compare the dynamics of the intruder and grains with packings of disks and pentagons. We observe that the packing of pentagons exerts comparable forces on the intruder as the packing of disks, though at significantly lower packing fractions. We also find from the average flow fields of grains during slip events that disks circulate around the intruder and rotate about their centers of mass significantly more than pentagons, which tend to flow forward from the intruder. Lastly, using photoelasticimetry, for the packing of disks we measure a significantly larger spatial extent of stresses around the annular cell, and a significantly larger fraction of events that feature back-bending force chains, compared with the packing of pentagons.

In the last set of experiments, we vary grain angularity of a vertical (gravity-packed) granular material sheared by a slider. We observe that the average shearing force required to initiate slip events increases with angularity. As a result, sticking periods last longer and slip events release more energy in packings with more angular grains. We also observe differences in the flow fields of disks and angular grains in slip events; disks tend to form a pile in front of the slider, while other grains do not. Moreover angular grains are able to form local column-like structures at the surface of the bed that prop up the slider during sticking periods, while disks do not. We lastly show that the depth of the shear band and the depth of stress fluctuations between sticking periods are unaffected by grain angularity.

Overall, these novel observations from each experiment demonstrate that friction and grain shape are important factors determining properties of macroscopic stick-slip dynamics of granular materials, stress transmission in stable granular materials, and grain-scale dynamics during slip events. Our observations also serve as motivation for more robust modeling and theoretical descriptions of granular stability and flow more generally by considering the influences of basal friction and changes in grain shape.

The physics of complex systems has grown considerably as a field in recent decades, largely due to improved computational technology and increased availability of systems level data. One area in which physics is of growing relevance is molecular biology. A new field, systems biology, investigates features of biological systems as a whole, a strategy of particular importance for understanding emergent properties that result from a complex network of interactions. Due to the complicated nature of the systems under study, the physics of complex systems has a significant role to play in elucidating the collective behavior.

In this dissertation, we explore three problems in the physics of complex systems, motivated in part by systems biology. The first of these concerns the applicability of Boolean models as an approximation of continuous systems. Studies of gene regulatory networks have employed both continuous and Boolean models to analyze the system dynamics, and the two have been found produce similar results in the cases analyzed. We ask whether or not Boolean models can generically reproduce the qualitative attractor dynamics of networks of continuously valued elements. Using a combination of analytical techniques and numerical simulations, we find that continuous networks exhibit two effects -- an asymmetry between on and off states, and a decaying memory of events in each element's inputs -- that are absent from synchronously updated Boolean models. We show that in simple loops these effects produce exactly the attractors that one would predict with an analysis of the stability of Boolean attractors, but in slightly more complicated topologies, they can destabilize solutions that are stable in the Boolean approximation, and can stabilize new attractors.

Second, we investigate ensembles of large, random networks. Of particular interest is the transition between ordered and disordered dynamics, which is well characterized in Boolean systems. Networks at the transition point, called critical, exhibit many of the features of regulatory networks, and recent studies suggest that some specific regulatory networks are indeed near-critical. We ask whether certain statistical measures of the ensemble behavior of large continuous networks are reproduced by Boolean models. We find that, in spite of the lack of correspondence between attractors observed in smaller systems, the statistical characterization given by the continuous and Boolean models show close agreement, and the transition between order and disorder known in Boolean systems can occur in continuous systems as well. One effect that is not present in Boolean systems, the failure of information to propagate down chains of elements of arbitrary length, is present in a class of continuous networks. In these systems, a modified Boolean theory that takes into account the collective effect of propagation failure on chains throughout the network gives a good description of the observed behavior. We find that propagation failure pushes the system toward greater order, resulting in a partial or complete suppression of the disordered phase.

Finally, we explore a dynamical process of direct biological relevance: asymmetric cell division in A. thaliana. The long term goal is to develop a model for the process that accurately accounts for both wild type and mutant behavior. To contribute to this endeavor, we use confocal microscopy to image roots in a SHORTROOT inducible mutant. We compute correlation functions between the locations of asymmetrically divided cells, and we construct stochastic models based on a few simple assumptions that accurately predict the non-zero correlations. Our result shows that intracellular processes alone cannot be responsible for the observed divisions, and that an intercell signaling mechanism could account for the measured correlations.

Limit-periodic (LP) structures exhibit a type of nonperiodic order yet to be found in a natural material. A recent result in tiling theory, however, has shown that LP order can spontaneously emerge in a two-dimensional (2D) lattice model with nearest-and next-nearest-neighbor interactions. In this dissertation, we explore the question of what types of interactions can lead to a LP state and address the issue of whether the formation of a LP structure in experiments is possible. We study emergence of LP order in three-dimensional (3D) tiling models and bring the subject into the physical realm by investigating systems with realistic Hamiltonians and low energy LP states. Finally, we present studies of the vibrational modes of a simple LP ball and spring model whose results indicate that LP materials would exhibit novel physical properties.

A 2D lattice model defined on a triangular lattice with nearest- and next-nearest-neighbor interactions based on the Taylor-Socolar (TS) monotile is known to have a LP ground state. The system reaches that state during a slow quench through an infinite sequence of phase transitions. Surprisingly, even when the strength of the next-nearest-neighbor interactions is zero, in which case there is a large degenerate class of both crystalline and LP ground states, a slow quench yields the LP state. The first study in this dissertation introduces 3D models closely related to the 2D models that exhibit LP phases. The particular 3D models were designed such that next-nearest-neighbor interactions of the TS type are implemented using only nearest-neighbor interactions. For one of the 3D models, we show that the phase transitions are first order, with equilibrium structures that can be more complex than in the 2D case.

In the second study, we investigate systems with physical Hamiltonians based on one of the 2D tiling models with the goal of stimulating attempts to create a LP structure in experiments. We explore physically realizable particle designs while being mindful of particular features that may make the assembly of a LP structure in an experimental system difficult. Through Monte Carlo (MC) simulations, we have found that one particle design in particular is a promising template for a physical particle; a 2D system of identical disks with embedded dipoles is observed to undergo the series of phase transitions which leads to the LP state.

LP structures are well ordered but nonperiodic, and hence have nontrivial vibrational modes. In the third section of this dissertation, we study a ball and spring model with a LP pattern of spring stiffnesses and identify a set of extended modes with arbitrarily low participation ratios, a situation that appears to be unique to LP systems. The balls that oscillate with large amplitude in these modes live on periodic nets with arbitrarily large lattice constants. By studying periodic approximants to the LP structure, we present numerical evidence for the existence of such modes, and we give a heuristic explanation of their structure.

Two-dimensional impact experiments by Clark et al. identified the source of inertial drag to be caused by ‘collisions’ with a latent force network, leading to large fluctuations of the force experienced by the impactor. These collisions provided the major drag on an impacting intruder until the intruder was nearly at rest. As a complement, we consider controlled pull-out experiments where a buried intruder is pulled out of a material, starting from rest. This provides a means to better understand the non-inertial part of the drag force, and to explore the mechanisms associated with the force fluctuations. The pull out process is a time reversed version of the impact process. In order to visualize this pulling process, we use 2D photoelastic disks from which circular intruders of different radii are pulled out. We check the effect of the initial depth of the intruder, as well as the widths and friction of boundaries. We present results about the dynamics of the intruder and the structures of the force chains inside the granular system as captured by high speed imaging. Before conducting the pull-out dynamic experiments, we first measured the critical pulling force that is needed to pull the intruder out. Under gradually increasing upward pulling force, a steadily strengthening force network forms in response to small displacements of intruder, then eventually fails and the intruder exits the material in a rapid event. We find that just before failure, the force chains bend in a way that is consistent with recent predictions by Blumenfeld and Ma. We found the boundary width together with friction plays an important role in this static pre-failure experiment. However, the system boundary does not have much effect on the dynamics of the intruder once the pull-out process starts.