The frictional pebble game: An algorithm for rigidity percolation in saturated frictional assemblies

A fast computer algorithm, the pebble game, has been used successfully to analyze the rigidity of two-dimensional (2D) elastic networks, as well as of a special class of 3D networks, the bond-bending networks, and enabled significant progress in studies of rigidity percolation on such networks. Application of the pebble game approach to general 3D networks has been hindered by the fact that the underlying mathematical theory is, strictly speaking, invalid in this case. We construct an approximate pebble game algorithm for general 3D networks, as well as a slower but exact algorithm, the relaxation algorithm, that we use for testing the new pebble game. Based on the results of these tests and additional considerations, we argue that in the particular case of randomly diluted central-force networks on bcc and fcc lattices, the pebble game is essentially exact. Using the pebble game, we observe an extremely sharp jump in the largest rigid cluster size in bond-diluted central-force networks in 3D, with the percolating cluster appearing and taking up most of the network after a single bond addition. This strongly suggests a first-order rigidity percolation transition, which is in contrast to the second-order transitions found previously for the 2D central-force and 3D bond-bending networks. While a first order rigidity transition has been observed previously for Bethe lattices and networks with "chemical order," here it is in a regular randomly diluted network. In the case of site dilution, the transition is also first order for bcc lattices, but results for fcc lattices suggest a second-order transition. Even in bond-diluted lattices, while the transition appears massively first order in the order parameter (the percolating cluster size), it is continuous in the elastic moduli. This, and the apparent nonuniversality, make this phase transition highly unusual.

Halogen bonded two-dimensional supramolecular assemblies studied by high resolution scanning tunneling microscopy

An Algorithm for Two-Dimensional Rigidity Percolation: The Pebble Game

The impact of electrostatic attraction on the uptake of gold nanoparticles (AuNPs) into positively charged strong poly-[2-(Methacryloyloxy) ethyl] trimethylammonium chloride (PMETAC) polyelectrolyte brushes was investigated. In this work, PMETAC brushes were synthesized via surface-initiated atom transfer radical polymerization (Si-ATRP). PMETAC/AuNP composite materials were prepared by incubation of the polymer brush coated samples into 3-mercaptopropionic acid-capped AuNP (5 nm in diameter) suspension. The electrostatic interactions were tuned by changing the surface charge of the AuNPs through variations in pH value, while the charge of the PMETAC brush was not affected. Atomic-force microscopy (AFM), ellipsometry, UV/Vis spectroscopy, gravimetric analysis and transmission electron microscopy (TEM) were employed to study the loading and penetration into the polymer brush. The results show that the number density of attached AuNPs depends on the pH value and increases with increasing pH value. There is also strong evidence that the particle assembly is dependent on the pH value of the AuNP suspension. Incubation of PMETAC brushes in AuNP suspension at pH 4 led to the formation of a surface layer on top of the brush (2D assembly) due to sterical hindrance of the clustered AuNPs, while incubation in AuNP suspension at pH 8 led to deeper particle penetration into the brush (3D assembly). The straightforward control of particle uptake and assembly by tuning the charge density of the nanoparticle surface is a valuable tool for the development of materials for colorimetric sensor applications.

The creation of shaped, uniform and colloidally stable two-dimensional (2D) assemblies by bottom-up methods represents a challenge of widespread current interest for a variety of applications. Herein, we describe the utilization of surface charge to stabilize self-assembled planar structures that are formed from crystallizable polymer precursors by a seeded growth approach. Addition of crystallizable homopolymers with charged end-groups to seeds generated by the sonication of block copolymer micelles with crystalline cores yields uniform platelet micelles with controlled dimensions. Significantly, the seeded growth approach is characterized by a morphological memory effect whereby the origin of the seed, which can involve a quasi-hexagonal or rectangular 2D platelet precursor, dictates the observed 2D platelet shape. This new strategy is illustrated using two different polymer systems, and opens the door to the construction of 2D hierarchical structures with broad utility.

Computer simulations were used to investigate shear flows of large numbers of viscoelastic, monosized, spherical particles in unbounded and bounded systems with solid fractions ranging from 0.16 to 0.59. A modified hard-sphere model with inelastic, instantaneous particle interactions was found to replicate some results predicted by kinetic theory in an unbounded shear flow at low and moderate solids fractions. This model was found to predict features such as particle lateral diffusive motion even for systems at solid fractions as high as 0.56. However, for higher solid fractions where phenomena such as jamming could occur, a particle dynamics model accounting for particle contacts of finite duration has been developed, in which the viscoelastic behavior of the particles was represented using a nonlinear Hertzian model. The nonlinear viscoelastic model was found to give more reasonable predictions for cluster formation than previously reported linear models, especially when accounting for surface friction in the model. However, neither frictionless nor frictional particle models could predict particle ordering in unbounded flows. As such, simulations were performed for bounded systems using both the modified hard-sphere model and the nonlinear particle dynamic model. For a bounded shear flow, particle ordering could be predicted by the hard-sphere model even in the absence of both particle friction and gravity, with the local solid fraction and wall separation distance governing the flow stability. For these conditions chain formation was found to be quite likely in the disordered layers for frictional particles. The interesting stick-slip dynamics could be clearly observed in the normal stress signal at the bottom wall. Interpretations were proposed for the complex processes observed, which could lay the foundation for further investigations in sheared dense granular systems.

Previous works have demonstrated that protein rigidity is related to thermodynamic stability, especially under conditions that favor formation of native structure. Mechanical network rigidity properties of a single conformation are efficiently calculated using the integer body-bar Pebble Game (PG) algorithm. However, thermodynamic properties require averaging over many samples from the ensemble of accessible conformations to accurately account for fluctuations in network topology. We have developed a mean field Virtual Pebble Game (VPG) that represents the ensemble of networks by a single effective network. That is, all possible number of distance constraints (or bars) that can form between a pair of rigid bodies is replaced by the average number. The resulting effective network is viewed as having weighted edges, where the weight of an edge quantifies its capacity to absorb degrees of freedom. The VPG is interpreted as a flow problem on this effective network, which eliminates the need to sample. Across a nonredundant dataset of 272 protein structures, we apply the VPG to proteins for the first time. Our results show numerically and visually that the rigidity characterizations of the VPG accurately reflect the ensemble averaged properties. This result positions the VPG as an efficient alternative to understand the mechanical role that chemical interactions play in maintaining protein stability.

Recent experimental measurements using twin-axis load cells in storage vessels for bulk solids have revealed load-dependent variation of the wall friction angles measured in the presence of small normal loads on smooth silo walls. The effect is quite appreciable with smooth particles such as plastic beads and glass ballotini, but particles with rough surfaces, like some agricultural seeds, show no such effect.The inclusion of an intrinsic shear resistance at zero normal load into the frictional behaviour (after Briscoe and Tabor) provides a satisfactory prediction of tangential forces required to cause gross sliding of smooth particles over smooth silo walls. This work applies the same adhesive friction concepts to the microdisplacements that occur before full tangential sliding develops.Mindlin analysed the traction profiles for two spheres in contact assuming a constant coefficient of friction, independent of normal stress inside the contact area, with microslip occurring in those portions of the contact area that had a tangential-to-normal stress ratio exceeding the friction coefficient. The assumptions of that analysis preclude the existence of the observed load dependence of wall friction angles. In this paper we re-examine the micromechanical nature of contacts between elastic spheres incorporating tangential stress limits that contain two terms: one proportional to the normal stress at each point in the contact area, the other independent of normal stress. Two possible physical (and mathematical) interpretations of the stress-independent term are discussed, one called adhesive slip and the other stick slip. The resulting traction profiles are examined to determine implications for both microslip and gross-sliding frictional behaviour. A complete theoretical analysis requires the solution of the surface displacement integral over the contact region for a chosen form of the tangential traction profile satisfying appropriate boundary conditions for elastic displacements. Upper and lower bounds and an approximate form for the resulting tangential force-displacement curves are developed and it is shown that such models result in a load-dependent friction behaviour that is very similar to recent experimental wall friction and individual particle friction measurements.

Control over the stacking patterns in 2D molecular assemblies is demonstrated using chemical modification. A target system is a hydrogen-bonded cocrystal (2:1) composed of 2-pyrrolidone (Py) and chloranilic acid (CA) (PyCA). X-ray crystallography showed that weak intersheet interactions give rise to a variety of metastable overlapping patterns comprised of the 2D assemblies mainly formed via hydrogen bonds, affording reversible and irreversible structural phase transitions. We prepared cocrystals of Py and anilic acids bearing different halogens, in which 2D assemblies isostructural with those observed in PyCA exhibit various overlapping patterns. The order of stability for each overlapping pattern estimated using calculations of the intermolecular interactions did not completely coincide with those indicated by our experimental results, which can be explained by considering the entropic effect: the molecular motion of Py as detected using nuclear quadrupole resonance spectroscopy.

Aromatic belts, ultrashort carbon nanotubes, and related structures, are emerging molecular entities in the fields of organic electronics and supramolecular chemistry owing to their structural rigidity, fully fused π-conjugation, and well-defined cavity. Synthesis of aromatic belts with embedded thiophene structures, which manifest significant optoelectronic and conductive properties, has not yet been achieved. Herein, we report the synthesis of thiophene-fused aromatic belts (thiophene belts) via one-step sulfur cross-linking reaction of partially fluorinated cycloparaphenylenes. Their structural features, including unidirectional columnar stacking with high dipole moment in crystals, two-dimensional layer assembly on metal surfaces, and photophysical properties, such as long-lifetime phosphorescence, are uncovered. These distinctive features of the thiophene belts should inspire a range of applications such as optoelectronic devices and polar materials.

A series of tetratopic C2-symmetric quasi-planar oligomeric phenylene−ethynylenes bearing carboxylic groups (OPE-COOHs), including the shortest 1,4-di((3,5-dicarboxyphenyl)ethynyl)benzene (TCB), midlength 4,4′-di((3,5-dicarboxyphenyl)ethynyl)biphenyl (TCBP), and the longest 4,4′′-di((3,5-dicarboxyphenyl)ethynyl)-p-terphenyl (TCTP), were synthesized via Pd(PPh3)4−CuI-catalyzed Sonogashira coupling reaction. Their molecular structures were characterized by nuclear magnetic resonance spectroscopy (NMR), matrix-assisted laser desorption−ionization time-of-flight mass spectroscopy (MALDI-TOF-MS), and element analyses. The molecule length effect of OPE-COOHs on two-dimensional (2D) assemblies at the octanoic acid−highly ordered pyrolytic graphite (HOPG) interface was investigated by scanning tunneling microscopy (STM) technique at ambient temperature and simulated by molecular modeling. The patterns and domain sizes of OPE-COOHs strongly depend on their molecule length with Kagomé network only for TCB and parallel network for TCBP and TCTP. TCTP with the longest rigid-rod OPE backbone exhibits clearly larger ordered domain size than that of TCB and TCBP. Besides, TCBP exhibits tight boundaries transition among different orientation domains via the acute or obtuse V-shaped chevron arrangements. Those different two-dimensional (2D) assembly behaviors will be favorable to get a further understanding of the condensed architectures in conjugated organic semiconductors.

The structure-directing effect of counterions with different shapes and charges has been investigated in the case of the complexes formed by uranyl ions with pimelic (heptanedioic) acid (H2C7) under solvo-hydrothermal conditions. Following the previously reported isolation of one-dimensional (1D) chains with the counterions [Fe(bipy)3]2+ and [Cu(phen)2]+, the complex [Ni(Me4phen)3]2[(UO2)4(C7)3(NO3)6]·CH3CN (1), with a bulkier cation, was synthesized and shown to consist of molecular anionic tetranuclear uranyl species, the presence of coordinated nitrate ions preventing polymerization. 1D zigzag chains are formed in [Ag(bipy)2]2[UO2(C7)(NO3)]2 (2), in which the cations, assembled into dimers through argentophilic interactions, are organized in columns through π-stacking interactions. The heterometallic complex [UO2Mn(C7)2(phen)2] (3) is formed in the presence of manganese(II) cations, and it crystallizes as a two-dimensional (2D) assembly in which manganese dimers display weak antiferromagnetic coupling ...

We report the design of two collagen-mimetic peptide sequences, NSI and NSII, that self-assemble into structurally defined nanoscale sheets. The underlying structure of these nanosheets can be understood in terms of the layered packing of collagen triple helices in two dimensions. These nanosheet assemblies represent a novel morphology for collagen-based materials, which, on the basis of their defined structure, may be envisioned as potentially biocompatible platforms for controlled presentation of chemical functionality at the nanoscale. The molecularly programmed self-assembly of peptides NSI and NSII into nanosheets suggests that sequence-specific macromolecules offer significant promise as design elements for two-dimensional (2D) assemblies. This investigation provides a design rubric for fabrication of structurally defined, peptide-based nanosheets using the principles of solution-based self-assembly facilitated through complementary electrostatic interactions.

We describe the two-dimensional (2D) assemblies of N,N'-dialkyl-substituted quinacridone derivatives on highly orientated pyrolytic graphite observed by scanning tunneling microscopy, and focus our discussion on whether the supramolecular organization can be modulated by the coadsorption of dicarboxylic acids. Our experiments have demonstrated that the quinacridone derivatives can form different 2D nanostructures when coadsorbed with dicarboxylic acids of different length at the liquid/graphite interface. Interestingly, N,N'-dihexadecyl-substituted quinacridone derivative alternately takes two different conformations in two columns for its coadsorption with pentadecanedioic acid and form a gridlike structure. It is shown that a cooperative effect of different interactions can be modulated by introducing guest molecule, leading to formation of different self-assembled nanostructures.

Statistical model of superconductivity in a 2D binary boson–fermion mixture