Nucleation-free 3D3D rigidity

When all non-edge distances of a graph realized in $\mathbb{R}^{d}$ as a {\em bar-and-joint framework} are generically {\em implied} by the bar (edge) lengths, the graph is said to be {\em rigid} in $\mathbb{R}^{d}$. For $d=3$, characterizing rigid graphs, determining implied non-edges and {\em dependent} edge sets remains an elusive, long-standing open problem. One obstacle is to determine when implied non-edges can exist without non-trivial rigid induced subgraphs, i.e., {\em nucleations}, and how to deal with them. In this paper, we give general inductive construction schemes and proof techniques to generate {\em nucleation-free graphs} (i.e., graphs without any nucleation) with implied non-edges. As a consequence, we obtain (a) dependent graphs in $3D$ that have no nucleation; and (b) $3D$ nucleation-free {\em rigidity circuits}, i.e., minimally dependent edge sets in $d=3$. It additionally follows that true rigidity is strictly stronger than a tractable approximation to rigidity given by Sitharam and Zhou \cite{sitharam:zhou:tractableADG:2004}, based on an inductive combinatorial characterization. As an independently interesting byproduct, we obtain a new inductive construction for independent graphs in $3D$. Currently, very few such inductive constructions are known, in contrast to $2D$

Spectral-element simulations of long-term fault slip: Effect of low-rigidity layers on earthquake-cycle dynamics

We develop a spectral element method for the simulation of long-term histories of spontaneous seismic and aseismic slip on faults subjected to tectonic loading. Our approach reproduces all stages of earthquake cycles: nucleation and propagation of earthquake rupture, postseismic slip and interseismic creep. We apply the developed methodology to study the effects of low-rigidity layers on the dynamics of the earthquake cycle in 2-D. We consider two cases: small (M ~ 1) earthquakes on a fault surrounded by a damaged fault zone and large (M ~ 7) earthquakes on a vertical strike-slip fault that cuts through shallow low-rigidity layers. Our results indicate how the source properties of repeating earthquakes are affected by the presence of a damaged fault zone with low rigidity. Compared to faults in homogeneous media, we find (1) reduction in the earthquake nucleation size, (2) amplification of slip rates during dynamic rupture propagation, (3) larger recurrence interval, and (4) smaller amount of aseismic slip. Based on linear stability analysis, we derive a theoretical estimate of the nucleation size as a function of the width and rigidity reduction of the fault zone layer, which is in good agreement with simulated nucleation sizes. We further examine the effects of vertically-stratified layers (e.g., sedimentary basins) on the nature of shallow coseismic slip deficit. Our results suggest that low-rigidity shallow layers alone do not lead to coseismic slip deficit. While the low-rigidity layers result in lower interseismic stress accumulation, they also cause dynamic amplification of slip rates, with the net effect on slip being nearly zero

Do thermodynamically stable rigid solids exist?

Customarily, crystalline solids are defined to be {\em rigid} since they resist changes of shape determined by their boundaries. However, rigid solids cannot exist in the thermodynamic limit where boundaries become irrelevant. Particles in the solid may rearrange to adjust to shape changes eliminating stress without destroying crystalline order. Rigidity is therefore valid only in the {\em metastable} state that emerges because these particle rearrangements in response to a deformation, or strain, are associated with slow collective processes. Here, we show that a thermodynamic collective variable may be used to quantify particle rearrangements that occur as a solid is deformed at zero strain rate. Advanced Monte Carlo simulation techniques are then employed to obtain the equilibrium free energy as a function of this variable. Our results lead to a new view on rigidity: While at zero strain a rigid crystal coexists with one that responds to infinitesimal strain by rearranging particles and expelling stress, at finite strain the rigid crystal is metastable, associated with a free energy barrier that decreases with increasing strain. The rigid phase becomes thermodynamically stable by switching on an external field, which penalises particle rearrangements. This produces a line of first-order phase transitions in the field - strain plane that intersects the origin. Failure of a solid once strained beyond its elastic limit is associated with kinetic decay processes of the metastable rigid crystal deformed with a finite strain rate. These processes can be understood in quantitative detail using our computed phase diagram as reference.Comment: 11 pages, 7 figure

Microscopic mechanism for experimentally observed anomalous elasticity of DNA in 2D

By exploring a recent model [Palmeri, J., M. Manghi, and N. Destainville. 2007. Phys. Rev. Lett. 99:088103] where DNA bending elasticity, described by the wormlike chain model, is coupled to base-pair denaturation, we demonstrate that small denaturation bubbles lead to anomalies in the flexibility of DNA at the nanometric scale, when confined in two dimensions (2D), as reported in atomic force microscopy (AFM) experiments [Wiggins, P. A., et al. 2006. Nature Nanotech. 1:137-141]. Our model yields very good fits to experimental data and quantitative predictions that can be tested experimentally. Although such anomalies exist when DNA fluctuates freely in three dimensions (3D), they are too weak to be detected. Interactions between bases in the helical double-stranded DNA are modified by electrostatic adsorption on a 2D substrate, which facilitates local denaturation. This work reconciles the apparent discrepancy between observed 2D and 3D DNA elastic properties and points out that conclusions about the 3D properties of DNA (and its companion proteins and enzymes) do not directly follow from 2D experiments by AFM.Comment: To appear in Biophys. J. 8 pages, supplementary information included (7 pages

Adhesion of membranes via receptor-ligand complexes: Domain formation, binding cooperativity, and active processes

Cell membranes interact via anchored receptor and ligand molecules. Central questions on cell adhesion concern the binding affinity of these membrane-anchored molecules, the mechanisms leading to the receptor-ligand domains observed during adhesion, and the role of cytoskeletal and other active processes. In this review, these questions are addressed from a theoretical perspective. We focus on models in which the membranes are described as elastic sheets, and the receptors and ligands as anchored molecules. In these models, the thermal membrane roughness on the nanometer scale leads to a cooperative binding of anchored receptor and ligand molecules, since the receptor-ligand binding smoothens out the membranes and facilitates the formation of additional bonds. Patterns of receptor domains observed in Monte Carlo simulations point towards a joint role of spontaneous and active processes in cell adhesion. The interactions mediated by the receptors and ligand molecules can be characterized by effective membrane adhesion potentials that depend on the concentrations and binding energies of the molecules.Comment: Review article, 13 pages, 9 figures, to appear in Soft Matte

Patterning porosity in hydrogels by arresting phase separation

Poly (ethylene glycol) (PEG) hydrogels have been used extensively in biological and tissue engineering, because of their outstanding biocompatibility and processability. However, it is not yet possible to process soft materials like PEG hydrogels with the requisite precision and throughput needed to recapitulate macroscopic biological tissue with control over every hierarchical scale. In this study, porous PEG hydrogels are processed by a phase separation method and patterned in a single photolithographic step. The thermodynamics of the temperature triggered spinodal decomposition of a ternary mixture of water, salt, and polymer are studied resulting in a ternary phase diagram and a spinodal temperature plot. Importantly, the state of porosity can be frozen by exposing the hydrogel to UV light to form a crosslinked hydrogel network. The average pore size can be tuned by changing delay between the application of heat and UV exposure. By utilizing grey-scale photomasks, a single process can be used to define regions of pure hydrogel, porous hydrogel with a programmed average pore size, and blank substrate with no hydrogel. In addition to representing a combination of a top-down and a bottom-up processes that enables the realization of complex samples, the simplicity of this process and the versatility of the resultant patterns could provide a useful capability for the definition of hydrogel samples for the development of advanced biomaterials

Crystal growth and elasticity

The purpose of this paper is to review some elasticity effects in epitaxial growth. We start by a description of the main ingredients needed to describe elasticity effects (elastic interactions, surface stress, bulk and surface elasticity, thermodynamics of stressed solids). Then we describe how bulk and surface elasticity affect growth mode and surface morphology by means of stress-driven instability. At last stress-strain evolution during crystal growth is reported.Comment: 12 page

Algorithms for detecting dependencies and rigid subsystems for CAD

Geometric constraint systems underly popular Computer Aided Design soft- ware. Automated approaches for detecting dependencies in a design are critical for developing robust solvers and providing informative user feedback, and we provide algorithms for two types of dependencies. First, we give a pebble game algorithm for detecting generic dependencies. Then, we focus on identifying the "special positions" of a design in which generically independent constraints become dependent. We present combinatorial algorithms for identifying subgraphs associated to factors of a particular polynomial, whose vanishing indicates a special position and resulting dependency. Further factoring in the Grassmann- Cayley algebra may allow a geometric interpretation giving conditions (e.g., "these two lines being parallel cause a dependency") determining the special position.Comment: 37 pages, 14 figures (v2 is an expanded version of an AGD'14 abstract based on v1