Comparison of rigidity and connectivity percolation in two dimensions

We show that the negative of the number of floppy modes behaves as a free energy for both connectivity and rigidity percolation, and we illustrate this result using Bethe lattices. The rigidity transition on Bethe lattices is found to be first order at a bond concentration close to that predicted by Maxwell constraint counting. We calculate the probability of a bond being on the infinite cluster and also on the overconstrained part of the infinite cluster, and show how a specific heatcan be defined as the second derivative of the free energy. We demonstrate that the Bethe lattice solution is equivalent to that of the random bond model, where points are joined randomly (with equal probability at all length scales) to have a given coordination, and then subsequently bonds are randomly removed.

Series Preface. Preface. Rigidity Theory. Generic and Abstract Rigidity B. Servatius, H. Servatius. Rigidity of Molecular Structures: Generic and Geometric Analysis W. Whiteley. Tensegrity Structures: Why Are They Stable? R. Connelly. The Role of Tensegrity in Distance Geometry T.F. Havel. Applications to Networks. Comparison of Connectivity and Rigidity Percolation C.F. Moukarzel, P.M. Duxbury. Rigidity Percolation on Trees P.L. Leath, C. Zeng. Rigidity as an Emergent Property of Random Networks: A Statistical Mechanical View P.M. Goldbart. Granular Matter Instability: A Structural Rigidity Point of View C.F. Moukarzel. Rigidity and Memory in a Simple Glass P. Chandra, L.B. Ioffe. Constraint Theory, Stiffness Percolation and the Rigidity Transition in Network Glasses J.C. Phillips. Topologically Disordered Networks of Rigid Polytopes: Applications to Noncrystalline Solids and Constrained Viscous Sintering P.K. Gupta. Rigidity Constraints in Amorphization of Singly- and Multiply-Polytopic Structures L.W. Hobbs, et al. Floppy Modes in Crystalline and Amorphous Silicates M.T. Dove, et al. Generic Rigidity of Networks Glasses M.F. Thorpe, et al. Rigidity Transition in Chalcogenide Glasses P. Boolchand, et al. Rigidity, Fragility, Bond Models and the `Energy Landscape' for Covalent Glassformers C.A. Angell. Entropic Rigidity B. Joos, et al. Applications to Proteins. Molecular Dynamics and Normal Mode Analysis of Biomolecular Rigidity D.A. Case. Efficient Stochastic Global Optimization for Protein Structure Prediction Y. Zhou, R. Abagyan. Flexible and Rigid Regions in Proteins D.J. Jacobs, et al. Flexibly Screening for Molecules Interacting with Proteins V. Schnecke, L.A. Kuhn. Studying Macromolecular Motions in a Database Framework: From Structure to Sequence M. Gerstein, et al. List of Participants. Index.

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

The evolution of wrinkles and folds in compressed thin films of type-purified single-wall carbon nanotubes (SWCNTs) on polydimethylsiloxane (PDMS) substrates is used to study the mechanical response of pristine nanotube networks. While the low-strain moduli are consistent with the exceptional mechanical properties of individual nanotubes, the films are remarkably fragile, exhibiting small yield strains that decrease with increasing thickness. We find significant differences in the mechanical response of semiconducting as compared to metallic SWCNT networks, and we use simple scaling arguments to relate these differences to previously determined Hamaker constants associated with each electronic type. A comparison with conductivity measurements performed on identical films suggests more than a two-fold variation in the onset of rigidity vs. connectivity percolation, and we discuss the potential implications of this for both rigid-rod networks and the use of type-purified SWCNTs in flexible electronics.

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.

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.

A note on generic rigidity of graphs in higher dimension

We have used the group theoretical method of Landau and Lifshitz to derive effective Hamiltonians for certain paramagnetic to antiferromagnetic transitions having order‐parameters with n?4 components, and we have performed a renormalization group analysis in 4−ε dimensions. The first‐order nature of the order‐disorder transitions in Cr(n=12), Eu(n=12), UO2(n=6), and MnO(n=8) can be explained by noting that the corresponding Hamiltonians possess no stable fixed points in 4−ε dimensions. We predict that all fcc type I(m̄⊥k̄), type II and type III(m̄⊥[100],k̄=[1/201]) antiferromagnetic transitions are first‐order. We hope our work will serve as a guide in an experimental search for new examples of first‐order transitions. We have also considered a 2m‐component Hamiltonian which possesses a unique, nonisotropic, stable fixed point for each value of 2m?4. When 2m=4, the Hamiltonian describes the paramagnetic to antiferromagnetic transitions in TbAu2, DyC2, Tb, Ho, Dy, and the structural transition in NbO2. If these transitions are second‐order, we predict they all belong to the same universality class. For 2m=6, the Hamiltonian describes the antiferromagnetic transitions in TbD2, Nd, K2IrCl6, and MnS2. We predict these transitions belong to a single universality class, and suggest that experiments be performed to test our results.

Rigidity percolation with g degrees of freedom per site is analyzed on randomly diluted Erdös-Renyi graphs, with average connectivity gamma, in the presence of a field h. In the (gamma,h) plane, the rigid and flexible phases are separated by a line of first-order transitions whose location is determined exactly. This line ends at a critical point with classical critical exponents. Analytic expressions are given for the densities n(F) of uncanceled degrees of freedom and gamma(r) of redundant bonds. Upon crossing the coexistence line, gamma(r) and n(F) are continuous, although their first derivatives are discontinuous. We extend, for the case of nonzero field, a recently proposed hypothesis, namely, that the density of uncanceled degrees of freedom is a "free energy" for rigidity percolation. Analytic expressions are obtained for the energy, entropy, and specific heat. Some analogies with a liquid-vapor transition are discussed. Particularizing to zero field, we find that the existence of a (g+1) core is a necessary condition for rigidity percolation with g degrees of freedom. At the transition point gamma(c), Maxwell counting of degrees of freedom is exact on the rigid cluster and on the (g+1) rigid core, i.e., the average coordination of these subgraphs is exactly 2g, although gamma(c), the average coordination of the whole system, is smaller than 2g. gamma(c) is found to converge to 2g for large g, i.e., in this limit Maxwell counting is exact globally as well.

Dense packings of particles of different size and shape are simulated by the mechanical contraction method (MCM). The effect of particle shape and particle size disparity on the packing density and micro-structure of two-component composite materials with competing length scales is investigated. We propose a simple analytical model to calculate the maximum packing fraction of such a mixture, which considers two co-existing random networks of large and small particles. The model predicts a non-monotonic behavior of the total packing fraction as a function of the mixture composition with the absolute maximum in packing fraction when the two sub-systems of large and small particles simultaneously jam. The mechanical contraction simulations corroborate this finding and also provide additional results for arbitrary particle size disparity. The simulated granular materials composed of frictionless spherical and rod-like particles also exhibit a non-monotonic dependence of the onset of rigidity percolation or jamming transition on the rod elongation.

A model of percolation with disaggregation and aggregation (PDA) is proposed and studied by means of the Monte Carlo simulation on the square lattice. The critical concentration φ c , the correlation length exponent and the fractal and spreading dimensions of the largest cluster at φ c have been computed. The obtained results suggest that the PDA model belongs to the same universality class than the standard percolation model. Nevertheless, the critical concentration (φ c =0.464±0.005) of the former is quite different from the critical probability (p c =0.5927) of the later model.

Quantum critical points in quasiperiodic magnets can realize new universality classes, with critical properties distinct from those of clean or disordered systems. Here, we study quantum phase transitions separating ferromagnetic and paramagnetic phases in the quasiperiodic q-state Potts model in 2+1D. Using a controlled real-space renormalization group approach, we find that the critical behavior is largely independent of q, and is controlled by an infinite-quasiperiodicity fixed point. The correlation length exponent is found to be ν=1, saturating a modified version of the Harris-Luck criterion.

Motivated by recent experiments on phase behavior of systems confined in porous media, we have studied the effect of randomness on the nature of the phase transition in the two-dimensional Potts model. To model the effects of the porous matrix we introduce a random distribution of couplings P(J(ij))=pdelta(J(ij)-J1)+(1-p)delta(J(ij)-J2) in the q state Potts Hamiltonian. An extensive Monte Carlo study is made on this system for q=5. We studied two different cases of disorder (a) J(1)/J(2)-->infinity and p=0.8 and (b) J(1)/J(2)=10 and p=0.5. We observed, in both cases, that the weak first order transition that appears in the pure case, changes to a second-order transition. A finite size scaling analysis shows that the correlation length exponent nu is close to 1 and the best fit to the dependence of the specific heat on system size is logarithmic. This suggests that both cases belong to the universality class of the Ising model. In contrast, the magnetic exponents point to a different universality class.

Renormalization group (RG) study of the Ising model on higher-dimensional ( d>1) quasi-lattices

Solid-state caloric effects, such as the magnetocaloric (MCE) and barocaloric (BCE) effects, may be utilized in future cooling technologies that are more efficient and environment-friendly. Large caloric effects often occur near phase transitions, especially near coupled first-order magnetostructural transitions (MST), and are initiated by external parameters, such as magnetic field or hydrostatic pressure. In this dissertation, the effects of pressure, temperature, and magnetic field on the phase transitions in three material systems are studied in order to elucidate how the respective caloric effects are affected. In the first study, the realization of a coupled MST in a MnNiSi-based system through isostructural alloying is explored, which resulted in a giant conventional MCE. The MST shifts towards lower temperature with increasing applied hydrostatic pressure, whereas it shifts towards higher temperature with an increase in magnetic field. The strong pressure dependence along with a large volume change during the MST suggested the possibility of pressure-induced BCE in this material. In a subsequent study, we observed a giant hydrostatic pressure induced inverse BCE through pressure-dependent calorimetric measurements. The multiple caloric effects in the same material for the same phase transition qualify this material as a multicaloric material. In second study, we investigated the hydrostatic pressure dependence of the metamagnetic transitions in DyRu2Si2, which shows multiple metamagnetic transitions at atmospheric pressure. With the application of moderate hydrostatic pressure, the metamagnetic transitions disappeared, but then reappeared with increasing pressure. We discuss the pressure-induced magnetostrictive behavior, the variation of the entropy changes with pressure, and a possible origin of the pressure-dependent behavior in light of the variation of the Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interactions. For x = 0.25 in the Ni2Mn1-xCuxGa Heusler alloy series, the structural and magnetic transitions coincide to create a coupled first-order MST. Since giant MCE was reported for this system, it is useful to understand the underlying physics driving the coupling of transitions. Although first-order transitions cannot be investigated through the critical behavior analysis, the structural and magnetic transitions in Ni2MnGa (parent alloy) and Ni2Mn0.85Cu0.15Ga are not coupled. In this case, investigating the critical behaviors of the two alloys near their second-order phase transitions will provide insight as to how the magnetism in these materials evolve with increasing copper doping. In this study, through the calculated critical exponent values, we identified the universality classes which best described the parent and Cu-doped (x = 0.15) alloys. The exponent values shed light into the range of