Rigidity Percolation Transition Research Articles

Gel-like mechanisms of durability and deformability in wet granular systems

It is known empirically that dry granular materials tend to crumble and that wetting them greatly increases their strength. Although the mechanism of the macroscopic material strength is known in homogeneously wetted granular system, the material strength in heterogeneously wetted granular system is not known due to the lack of experimental studies. Here, we focus on sand grains coated with silicone oil, whose wettability is stable with respect to time, and constructed a model system that can control the heterogeneity of interaction by mixing coated sand grains. The results show that the rapid increase in Young’s modulus is due to a rigidity percolation transition, and that the recombination of the network makes the material more resistant to deformation. This system leads to understanding the properties of jamming systems such as glasses, emulsions, and foams, where the effects of attractive interactions and rigidity percolation have been still unclear.

Multifunctional twisted kagome lattices: Tuning by pruning mechanical metamaterials.

This article investigates phonons and elastic response in randomly diluted lattices constructed by combining (via the addition of next-nearest bonds) a twisted kagome lattice, with bulk modulus B=0 and shear modulus G>0, with either a generalized untwisted kagome lattice with B>0 and G>0 or with a honeycomb lattice with B>0 and G=0. These lattices exhibit jamming-like critical endpoints at which B,G, or both B and G jump discontinuously from zero while the remaining moduli (if any) begin to grow continuously from zero. Pairs of these jamming points are joined by lines of continuous rigidity percolation transitions at which both B and G begin to grow continuously from zero. The Poisson ratio and G/B can be continuously tuned throughout their physical range via random dilution in a manner analogous to "tuning by pruning" in random jammed lattices. These lattices can be produced with modern techniques, such as three-dimensional printing, for constructing metamaterials.

Ultrafast light-induced softening of chalcogenide thin films above the rigidity percolation transition

Little is known about the role of network rigidity in light-induced structural rearrangements in network glasses due to a lack of supporting experiments and theories. In this article, we demonstrate for the first time the ultrafast structural rearrangements manifested as induced absorption (IA) over a broad spectral range in a-GexAs35-xSe65 thin films above the mean-field rigidity percolation transition, quantified by the mean coordination number ⟨r⟩ = 2.40. The IA spectrum arising from self-trapped excitons induced structural rearrangements by softening the glass network that strikingly reveals two relaxation mechanisms which differ by one order of magnitude. The fast kinetics of electron-lattice interaction occurs within 1 ps, exhibits a weak dependence on rigidity, and dominates in the sub-bandgap region. In a stark contrast, the slow kinetics is associated with the structural changes in the bandgap region and depends strongly on network rigidity. Our results further demonstrate that amplitude of IA scales a linear relationship with excitation fluence which provides a unique way to induce structural rearrangements in an over-coordinated network to exploit it for practical purposes. Our results modify the conventional concept of rigidity dependence of light-induced effects in network glasses, when excited with an ultrafast laser.

Elasticity of randomly diluted honeycomb and diamond lattices with bending forces

We use numerical simulations and an effective-medium theory to study the rigidity percolation transition of the honeycomb and diamond lattices when weak bond-bending forces are included. We use a rotationally invariant bond-bending potential, which, in contrast to the Keating potential, does not involve any stretching. As a result, the bulk modulus does not depend on the bending stiffness κ. We obtain scaling functions for the behavior of some elastic moduli in the limits of small , and small , where is an occupation probability of each bond, and is the critical probability at which rigidity percolation occurs. We find good quantitative agreement between effective-medium theory and simulations for both lattices for close to one.

Rigidity percolation by next-nearest-neighbor bonds on generic and regular isostatic lattices.

We study rigidity percolation transitions in two-dimensional central-force isostatic lattices, including the square and the kagome lattices, as next-nearest-neighbor bonds ("braces") are randomly added to the system. In particular, we focus on the differences between regular lattices, which are perfectly periodic, and generic lattices with the same topology of bonds but whose sites are at random positions in space. We find that the regular square and kagome lattices exhibit a rigidity percolation transition when the number of braces is ∼LlnL, where L is the linear size of the lattice. This transition exhibits features of both first-order and second-order transitions: The whole lattice becomes rigid at the transition, and a diverging length scale also exists. In contrast, we find that the rigidity percolation transition in the generic lattices occur when the number of braces is very close to the number obtained from Maxwell's law for floppy modes, which is ∼L. The transition in generic lattices is a very sharp first-order-like transition, at which the addition of one brace connects all small rigid regions in the bulk of the lattice, leaving only floppy modes on the edge. We characterize these transitions using numerical simulations and develop analytic theories capturing each transition. Our results relate to other interesting problems, including jamming and bootstrap percolation.

Rheological properties of graphene oxide liquid crystal

We report the rheological properties of liquid crystalline graphene oxide (GO) aqueous dispersion. GO dispersions exhibit typical shear thinning behaviors of liquid crystals, which is described by power law or simple Curreau model. Irrespective of the shear rate, shear viscosity exhibits sudden decrease with the increase of GO composition around a critical volume fraction, ϕc=0.33%, demonstrating typical colloidal isotropic–nematic phase transition. Dynamic measurements reveal the liquid-like (isotropic phase, G′>G″) behavior at a low GO composition (ϕ∼0.08%) and solid-like (liquid crystalline) behavior at higher compositions (ϕ∼0.45%), where G′ exceeds over G″. Nematic gel-like phase is confirmed at a higher GO composition over ϕ>0.83%, where both G′ and G″ moduli are nearly independent of frequency (ω). Simple power law scaling arguments are introduced to model the dependence of yield stress and viscoelastic moduli on the GO composition. We also observed the yield stress and rigidity percolation transition above phase transition composition ϕc>0.33% with a percolation exponent of 1.3±0.1. These rheological insights provide valuable information for the liquid crystalline processing of GO based materials including fibers, sheets and other complex structures for electronic/optoelectronic and energy storage/conversion applications.

Two rigidity-percolation transitions on binary Bethe networks and the intermediate phase in glass

Rigidity percolation is studied analytically on randomly bonded networks with two types of nodes, respectively, with coordination numbers z(1) and z(2), and with g(1) and g(2) degrees of freedom each. For certain cases that model chalcogenide glass networks, two transitions, both of first order, are found, with the first transition usually rather weak. The ensuing intermediate pase, although not isostatic in its entirety, has very low self-stress. Our results suggest a possible mechanism for the appearance of intermediate phases in glass that does not depend on a self-organization principle.

Elasticity of a filamentous kagome lattice

The diluted kagome lattice, in which bonds are randomly removed with probability 1-p, consists of straight lines that intersect at points with a maximum coordination number of 4. If lines are treated as semiflexible polymers and crossing points are treated as cross-links, this lattice provides a simple model for two-dimensional filamentous networks. Lattice-based effective-medium theories and numerical simulations for filaments modeled as elastic rods, with stretching modulus μ and bending modulus κ, are used to study the elasticity of this lattice as functions of p and κ. At p=1, elastic response is purely affine, and the macroscopic elastic modulus G is independent of κ. When κ=0, the lattice undergoes a first-order rigidity-percolation transition at p=1. When κ>0, G decreases continuously as p decreases below one, reaching zero at a continuous rigidity-percolation transition at p=p(b)≈0.605 that is the same for all nonzero values of κ. The effective-medium theories predict scaling forms for G, which exhibit crossover from bending-dominated response at small κ/μ to stretching-dominated response at large κ/μ near both p=1 and p(b), that match simulations with no adjustable parameters near p=1. The affine response as p→1 is identified with the approach to a state with sample-crossing straight filaments treated as elastic rods.

Yield stress, thixotropy and shear banding in a dilute aqueous suspension of few layer graphene oxide platelets

We demonstrate a rigidity percolation transition and the onset of yield stress in a dilute aqueous dispersion of graphene oxide platelets (aspect ratio ∼5000) above a critical volume fraction of 3.75 × 10−4 with a percolation exponent of 2.4 ± 0.1. The viscoelastic moduli of the gel at rest measured as a function of time indicate the absence of structural evolution of the 3D percolated network of disks. However a shear-induced aging giving rise to a compact jammed state and shear rejuvenation indicating a homogenous flow is observed when a steady shear stress (σ) is imposed in creep experiments. We construct a shear diagram (σ vs. volume fraction ϕ) and the critical stress above which shear rejuvenation occurs is identified as the yield stress σy of the gel. The minimum steady state shear rate m obtained from creep experiments agrees well with the end of the plateau region in a controlled shear rate flow curve, indicating a shear localization below m. A steady state shear banding in the plateau region of the flow curve observed in particle velocimetry measurements in a Couette geometry confirms that the dilute suspensions of GO platelets form a thixotropic yield stress fluid.

Algorithms for three-dimensional rigidity analysis and a first-order percolation transition

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.

Exactly Solvable Models of Adaptive Networks

A satisfiability-unsatisfiability (SAT-UNSAT) transition takes place for many optimization problems when the number of constraints, graphically represented by links between variables nodes, is brought above some threshold. If the network of constraints is allowed to adapt by redistributing its links, the SAT-UNSAT transition may be delayed and preceded by an intermediate phase where the structure self-organizes to satisfy the constraints. We present an analytic approach, based on the recently introduced cavity method for large deviations, which exactly describes the two phase transitions delimiting this adaptive intermediate phase. We give explicit results for random bond models subject to the connectivity or rigidity percolation transitions, and compare them with numerical simulations.

Giant Photoplastic Effect in Vitreous Semiconductors near the Rigidity Percolation Transition

The negative giant photoplastic effect (giant photosoftening) has been experimentally observed in the As-Se system when films obtained by thermal evaporation of As20Se80 chalcogenide glass are irradiated by light from the region of the fundamental absorption edge. Correlation has been found between the photoplastic effect and rigidity percolation transition in the As-Se chalcogenide glass matrix. Such a correlation is not revealed when light irradiation changes the optical properties of these glass films. It has been shown that a nonlinear (non-Hookian) mechanism of the formation of the strain response is realized in the films subjected to the combined action of light and external mechanical loading.

Viscoelasticity of Single Wall Carbon Nanotube Suspensions

We investigate the viscoelastic properties of an associating rigid rod network: aqueous suspensions of surfactant stabilized single wall carbon nanotubes (SWNTs). The SWNT suspensions exhibit a rigidity percolation transition with an onset of solidlike elasticity at a volume fraction of 0.0026; the percolation exponent is 2.3+/-0.1. At large strain, the solidlike samples show volume fraction dependent yielding. We develop a simple model to understand these rheological responses and show that the shear dependent stresses can be scaled onto a single master curve to obtain an internanotube interaction energy per bond approximately 40k(B)T. Our experimental observations suggest SWNTs in suspension form interconnected networks with bonds that freely rotate and resist stretching. Suspension elasticity originates from bonds between SWNTs rather than from the stiffness or stretching of individual SWNTs.

Structural transitions in aqueous suspensions of natural graphite

The electric conductivity σ and viscosity η of aqueous suspensions at different volume fraction of graphite ϕ and concentration of nonionic surfactant (Triton X-305) are investigated. The correlations between conductivity and rheological properties are discussed. A model of structural transitions in aqueous graphite suspensions is discussed. Two structural transitions, corresponding to changes in electrical and rheomechanical properties, are identified as electrical connectivity and rigidity (or sol–gel) percolation transitions, respectively. An unusual initial decrease of the bulk electrical conductivity with graphite volume fraction ϕ increase was observed in the surfactant solution. This fact is explained as a result of an isolating coating formation around the graphite particles.

Light-induced giant softening of network glasses observed near the mean-field rigidity transition.

The longitudinal acoustic (LA) mode of bulk GexSe1-x glasses is examined in Brillouin scattering (BS) over the 0.15<x<1/3 range using near band-gap radiation (lambda=647.1 nm). The LA-mode frequency (nu(LA)) softens with increasing laser power in an athermal and reversible manner by nearly 30% (=Delta nu(LA)/nu(LA)) near x=x(c)=0.19(1) or mean coordination number, r(c)=2+2x(c)=2.38(2), close to the mean-field rigidity percolation transition (r(t)=2.40). BS is a bulk probe of elasticity, and photosoftening is maximized here when network stress is minimized near the elastic phase transition.

Nonuniversality of elastic exponents in random bond-bending networks.

We numerically investigate the rigidity percolation transition in two-dimensional flexible, random rod networks with freely rotating cross links. Near the transition, networks are dominated by bending modes and the elastic modulii vanish with an exponent f=3.0+/-0.2, in contrast with central force percolation which shares the same geometric exponents. This indicates that universality for geometric quantities does not imply universality for elastic ones. The implications of this result for actin-fiber networks is discussed.

Directed rigidity and bootstrap percolation in 1+1 dimensions.

We study directed rigidity percolation (equivalent to directed bootstrap percolation) on three different lattices: square, triangular, and augmented triangular. The first two of these display a first-order transition at p=1, while the augmented triangular lattice shows a continuous transition at a nontrivial p(c). On the augmented triangular lattice we find, by extensive numerical simulation, that the the directed rigidity percolation transition belongs to the same universality class as the directed percolation. The same conclusion is reached by studying its surface critical behavior, i.e., the spreading of rigidity from finite clusters close to a nonrigid wall. Near the discontinuous transition at p=1 on the triangular lattice, we are able to calculate the finite-size behavior of the density of rigid sites analytically. Our results are confirmed by numerical simulation.

Comparison of rigidity and connectivity percolation in two dimensions

Using a recently developed algorithm for generic rigidity of two-dimensional graphs, we analyze rigidity and connectivity percolation transitions in two dimensions on lattices of linear size up to $L=4096.$ We compare three different universality classes: the generic rigidity class, the connectivity class, and the generic ``braced square net''(GBSN). We analyze the spanning cluster density ${P}_{\ensuremath{\infty}},$ the backbone density ${P}_{B}$, and the density of dangling ends ${P}_{D}.$ In the generic rigidity (GR) and connectivity cases, the load-carrying component of the spanning cluster, the backbone, is fractal at ${p}_{c},$ so that the backbone density behaves as $B\ensuremath{\sim}(p\ensuremath{-}{p}_{c}{)}^{{\ensuremath{\beta}}^{\ensuremath{'}}}$ for $p&gt;{p}_{c}.$ We estimate ${\ensuremath{\beta}}_{\mathrm{gr}}^{\ensuremath{'}}=0.25\ifmmode\pm\else\textpm\fi{}0.02$ for generic rigidity and ${\ensuremath{\beta}}_{c}^{\ensuremath{'}}=0.467\ifmmode\pm\else\textpm\fi{}0.007$ for the connectivity case. We find the correlation length exponents ${\ensuremath{\nu}}_{\mathrm{gr}}=1.16\ifmmode\pm\else\textpm\fi{}0.03$ for generic rigidity compared to the exact value for connectivity, ${\ensuremath{\nu}}_{c}=\frac{4}{3}.$ In contrast the GBSN undergoes a first-order rigidity transition, with the backbone density being extensive at ${p}_{c},$ and undergoing a jump discontinuity on reducing p across the transition. We define a model which tunes continuously between the GBSN and GR classes, and show that the GR class is typical.

Duxbury, Moukarzel and Leath Reply:

Reply to a comment on "Infinite-Cluster geometry in central-force networks", PRL 78 (1997), 1480. A discussion about the order of the rigidity percolation transition.

Generic rigidity percolation in two dimensions.

We study rigidity percolation for random central-force networks on the bondand site-diluted generic triangular lattice. Here, each site location is randomly displaced from the perfect lattice, removing any special symmetries. Using the pebble game algorithm, the total number of floppy modes are counted exactly, and exhibit a cusp singularity in the second derivative at the transition from a rigid to a floppy structure. The critical thresholds for bond and site dilution are found to be 0.66020\ifmmode\pm\else\textpm\fi{}0.0003 and 0.69755\ifmmode\pm\else\textpm\fi{}0.0003, respectively. The network is decomposed into unique rigid clusters, and we apply the usual percolation scaling theory. From finite size scaling, we find that the generic rigidity percolation transition is second order, but in a different universality class from connectivity percolation, with the exponents \ensuremath{\alpha}=-0.48\ifmmode\pm\else\textpm\fi{}0.05, \ensuremath{\beta}=0.175\ifmmode\pm\else\textpm\fi{}0.02, and \ensuremath{\nu}=1.21\ifmmode\pm\else\textpm\fi{}0.06. The fractal dimension of the spanning rigid clusters and the spanning stressed regions at the critical threshold are found to be ${\mathit{d}}_{\mathit{f}}$=1.86\ifmmode\pm\else\textpm\fi{}0.02 and ${\mathit{d}}_{\mathit{BB}}$=1.80\ifmmode\pm\else\textpm\fi{}0.03, respectively. \textcopyright{} 1996 The American Physical Society.