Stabilization of nonlinear velocity profiles in athermal systems undergoing planar shear flow

We perform molecular dynamics simulations of model granular systems undergoing boundary-driven planar shear flow in two spatial dimensions with the goal of developing a more complete understanding of how dense particulate systems respond to applied shear. In particular, we are interested in determining when these systems will possess linear velocity profiles and when they will develop highly localized velocity profiles in response to shear. In previous work on similar systems we showed that nonlinear velocity profiles form when the speed of the shearing boundary exceeds the speed of shear waves in the material. However, we find that nonlinear velocity profiles in these systems are unstable at very long times. The degree of nonlinearity slowly decreases in time; the velocity profiles become linear when the granular temperature and density profiles are uniform across the system at long times. We measure the time $t_l$ required for the velocity profiles to become linear and find that $t_l$ increases as a power-law with the speed of the shearing boundary and increases rapidly as the packing fraction approaches random close packing. We also performed simulations in which differences in the granular temperature across the system were maintained by vertically vibrating one of the boundaries during shear flow. We find that nonlinear velocity profiles form and are stable at long times if the difference in the granular temperature across the system exceeds a threshold value that is comparable to the glass transition temperature in an equilibrium system at the same average density. Finally, the sheared and vibrated systems form stable shear bands, or highly localized velocity profiles, when the applied shear stress is lowered below the yield stress of the static part of the system.Comment: 11 pages, 14 figure

Tuning Jammed Frictionless Disk Packings from Isostatic to Hyperstatic

We perform extensive computational studies of two-dimensional static bidisperse disk packings using two distinct packing-generation protocols. The first involves thermally quenching equilibrated liquid configurations to zero temperature over a range of thermal quench rates $r$ and initial packing fractions followed by compression and decompression in small steps to reach packing fractions $\phi_J$ at jamming onset. For the second, we seed the system with initial configurations that promote micro- and macrophase-separated packings followed by compression and decompression to $\phi_J$. We find that amorphous, isostatic packings exist over a finite range of packing fractions from $\phi_{\rm min} \le \phi_J \le \phi_{\rm max}$ in the large-system limit, with $\phi_{\rm max} \approx 0.853$. In agreement with previous calculations, we obtain $\phi_{\rm min} \approx 0.84$ for $r > r^*$, where $r^*$ is the rate above which $\phi_J$ is insensitive to rate. We further compare the structural and mechanical properties of isostatic versus hyperstatic packings. The structural characterizations include the contact number, bond orientational order, and mixing ratios of the large and small particles. We find that the isostatic packings are positionally and compositionally disordered, whereas bond-orientational and compositional order increase with contact number for hyperstatic packings. In addition, we calculate the static shear modulus and normal mode frequencies of the static packings to understand the extent to which the mechanical properties of amorphous, isostatic packings are different from partially ordered packings. We find that the mechanical properties of the packings change continuously as the contact number increases from isostatic to hyperstatic.Comment: 11 pages, 15 figure

Entropy and Temperature of a Static Granular Assembly

Granular matter is comprised of a large number of particles whose collective behavior determines macroscopic properties such as flow and mechanical strength. A comprehensive theory of the properties of granular matter, therefore, requires a statistical framework. In molecular matter, equilibrium statistical mechanics, which is founded on the principle of conservation of energy, provides this framework. Grains, however, are small but macroscopic objects whose interactions are dissipative since energy can be lost through excitations of the internal degrees of freedom. In this work, we construct a statistical framework for static, mechanically stable packings of grains, which parallels that of equilibrium statistical mechanics but with conservation of energy replaced by the conservation of a function related to the mechanical stress tensor. Our analysis demonstrates the existence of a state function that has all the attributes of entropy. In particular, maximizing this state function leads to a well-defined granular temperature for these systems. Predictions of the ensemble are verified against simulated packings of frictionless, deformable disks. Our demonstration that a statistical ensemble can be constructed through the identification of conserved quantities other than energy is a new approach that is expected to open up avenues for statistical descriptions of other non-equilibrium systems.Comment: 5 pages, 4 figure

The [4+2]‐Cycloaddition of α‐Nitrosoalkenes with Thiochalcones as a Prototype of Periselective Hetero‐Diels–Alder Reactions—Experimental and Computational Studies

The [4+2]‐cycloadditions of α‐nitrosoalkenes with thiochalcones occur with high selectivity at the thioketone moiety of the dienophile providing styryl‐substituted 4H‐1,5,2‐oxathiazines in moderate to good yields. Of the eight conceivable hetero‐Diels–Alder adducts only this isomer was observed, thus a prototype of a highly periselective and regioselective cycloaddition has been identified. Analysis of crude product mixtures revealed that the α‐nitrosoalkene also adds competitively to the thioketone moiety of the thiochalcone dimer affording bis‐heterocyclic [4+2]‐cycloadducts. The experiments are supported by high‐level DFT calculations that were also extended to related hetero‐Diels–Alder reactions of other nitroso compounds and thioketones. These calculations reveal that the title cycloadditions are kinetically controlled processes confirming the role of thioketones as superdienophiles. The computational study was also applied to the experimentally studied thiochalcone dimerization, and showed that the 1,2‐dithiin and 2H‐thiopyran isomers are in equilibrium with the monomer. Again, the DFT calculations indicate kinetic control of this process

Geometrical families of mechanically stable granular packings

We enumerate and classify nearly all of the possible mechanically stable (MS) packings of bidipserse mixtures of frictionless disks in small sheared systems. We find that MS packings form continuous geometrical families, where each family is defined by its particular network of particle contacts. We also monitor the dynamics of MS packings along geometrical families by applying quasistatic simple shear strain at zero pressure. For small numbers of particles (N < 16), we find that the dynamics is deterministic and highly contracting. That is, if the system is initialized in a MS packing at a given shear strain, it will quickly lock into a periodic orbit at subsequent shear strain, and therefore sample only a very small fraction of the possible MS packings in steady state. In studies with N>16, we observe an increase in the period and random splittings of the trajectories caused by bifurcations in configuration space. We argue that the ratio of the splitting and contraction rates in large systems will determine the distribution of MS-packing geometrical families visited in steady-state. This work is part of our long-term research program to develop a master-equation formalism to describe macroscopic slowly driven granular systems in terms of collections of small subsystems.Comment: 18 pages, 23 figures, 5 table

Reconciling Semiclassical and Bohmian Mechanics: II. Scattering states for discontinuous potentials

In a previous paper [J. Chem. Phys. 121 4501 (2004)] a unique bipolar decomposition, Psi = Psi1 + Psi2 was presented for stationary bound states Psi of the one-dimensional Schroedinger equation, such that the components Psi1 and Psi2 approach their semiclassical WKB analogs in the large action limit. Moreover, by applying the Madelung-Bohm ansatz to the components rather than to Psi itself, the resultant bipolar Bohmian mechanical formulation satisfies the correspondence principle. As a result, the bipolar quantum trajectories are classical-like and well-behaved, even when Psi has many nodes, or is wildly oscillatory. In this paper, the previous decomposition scheme is modified in order to achieve the same desirable properties for stationary scattering states. Discontinuous potential systems are considered (hard wall, step, square barrier/well), for which the bipolar quantum potential is found to be zero everywhere, except at the discontinuities. This approach leads to an exact numerical method for computing stationary scattering states of any desired boundary conditions, and reflection and transmission probabilities. The continuous potential case will be considered in a future publication.Comment: 18 pages, 8 figure

Characterization and cloning of fasciclin I and fasciclin II glycoproteins in the grasshopper

Monoclonal antibodies were previously used to identify two glycoproteins, called fasciclin I and II (70 and 95 kDa, respectively), which are expressed on different subsets of axon fascicles in the grasshopper (Schistocerca americana) embryo. Here the monoclonal antibodies were used to purify these two membrane-associated glycoproteins for further characterization. Fasciclin II appears to be an integral membrane protein, where fasciclin I is an extrinsic membrane protein. The amino acid sequences of the amino terminus and fragments of both proteins were determined. Using synthetic oligonucleotide probes and antibody screening, we isolated genomic and cDNA clones. Partial DNA sequences of these clones indicate that they encode fasciclins I and II