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Arbabi S, Sahimi M. Elastic moduli of body-centered cubic lattice near rigidity percolation threshold: Finite-size effects and evidence for first-order phase transition. Phys Rev E 2021; 103:042314. [PMID: 34005887 DOI: 10.1103/physreve.103.042314] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/10/2020] [Accepted: 04/02/2021] [Indexed: 11/07/2022]
Abstract
Extensive numerical simulations of rigidity percolation with only central forces in large three-dimensional lattices have indicated that many of their topological properties undergo a first-order phase transition at the rigidity percolation threshold p_{ce}. In contrast with such properties, past numerical calculations of the elastic moduli of the same lattices had provided evidence for a second-order phase transition. In this paper we present the results of extensive simulation of rigidity percolation in large body-centered cubic (bcc) lattices, and show that as the linear size L of the lattice increases, the elastic moduli close to p_{ce} decrease in a stepwise, discontinuous manner, a feature that is absent in lattices with L<30. The number and size of such steps increase with L. As p_{ce} is approached, long-range, nondecaying orientational correlations are built up, giving rise to compact, nonfractal clusters. As a result, we find that the backbone of the lattice at p_{ce} is compact with a fractal dimension D_{bb}≈3. The absence of fractal, scale-invariant clusters, the hallmark of second-order phase transitions, together with the stairwise behavior of the elastic moduli, provide strong evidence that, at least in bcc lattices, many of the topological properties of rigidity percolation as well as its elastic moduli may undergo a first-order phase transition at p_{ce}. In relatively small lattices, however, the boundary effects interfere with the nonlocal nature of the rigidity percolation. As a result, only when such effects diminish in large lattices does the true nature of the phase transition emerge.
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Affiliation(s)
- Sepehr Arbabi
- Department of Chemical Engineering, The University of Texas Permian Basin, Odessa, Texas 79762, USA
| | - Muhammad Sahimi
- Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, California 90089-1211, USA
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Ashton DJ, Garrahan JP. Relationship between vibrations and dynamical heterogeneity in a model glass former: extended soft modes but local relaxation. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2009; 30:303-307. [PMID: 19859750 DOI: 10.1140/epje/i2009-10531-6] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/24/2008] [Revised: 07/31/2009] [Accepted: 10/08/2009] [Indexed: 05/28/2023]
Abstract
We study the relation between short-time vibrational modes and long-time relaxational dynamics in a kinetically constrained lattice gas with harmonic interactions between neighbouring particles. We find a correlation between the location of the low- (high-) frequency vibrational modes and regions of high (low) propensity for motion. This is similar to what was observed in continuous force systems, but our interpretation is different: in our case relaxation is due to localised excitations which propagate through the system; these localised excitations act as background disorder for the elastic network, giving rise to anomalous vibrational modes. Our results provide an example whereby a correlation between spatially extended low-frequency modes and high-propensity regions does not imply that relaxational dynamics originates in extended soft modes but rather belies their common origin. We consider other measures of elastic heterogeneity, such as non-affine displacement fields and mode localisation lengths, and discuss implications of our results to interpretations of dynamic heterogeneity more generally.
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Affiliation(s)
- D J Ashton
- School of Physics and Astronomy, University of Nottingham, NG72RD Nottingham, UK.
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Abstract
The ability of materials to self-heal from mechanical and thermally induced damage is explored in this paper and has significance in the field of fracture and fatigue. The history and evolution of several self-repair systems is examined including nano-beam healing elements, passive self-healing, autonomic self-healing and ballistic self-repair. Self-healing mechanisms utilized in the design of these unusual materials draw much information from the related field of polymer-polymer interfaces and crack healing. The relationship of material damage to material healing is examined in a manner to provide an understanding of the kinetics and damage reversal processes necessary to impart self-healing characteristics. In self-healing systems, there are transitions from hard-to-soft matter in ballistic impact and solvent bonding and conversely, soft-to-hard matter transitions in high rate yielding materials and shear-thickening fluids. These transitions are examined in terms of a new theory of the glass transition and yielding, viz., the twinkling fractal theory of the hard-to-soft matter transition. Success in the design of self-healing materials has important consequences for material safety, product performance and enhanced fatigue lifetime.
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Affiliation(s)
- Richard P Wool
- Department of Chemical Engineering, University of Delaware, Newark DE 19716-3144, USA.
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Chubynsky MV, Thorpe MF. Algorithms for three-dimensional rigidity analysis and a first-order percolation transition. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:041135. [PMID: 17994964 DOI: 10.1103/physreve.76.041135] [Citation(s) in RCA: 35] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/27/2007] [Indexed: 05/25/2023]
Abstract
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.
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Affiliation(s)
- M V Chubynsky
- Département de Physique, Université de Montréal, Case Postale 6128, Succursale Centre-Ville, Montréal, Québec, Canada H3C 3J7.
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Moukarzel CF. Rigidity percolation in a field. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:056104. [PMID: 14682843 DOI: 10.1103/physreve.68.056104] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/14/2003] [Indexed: 05/24/2023]
Abstract
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.
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Affiliation(s)
- Cristian F Moukarzel
- Departmento de Física Aplicada, CINVESTAV del IPN, Avenida Tecnológico Km 6, 97310 Mérida, Yucatán, Mexico.
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Zhou Z, Joós B, Lai PY. Elasticity of randomly diluted central force networks under tension. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:055101. [PMID: 14682830 DOI: 10.1103/physreve.68.055101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/02/2003] [Indexed: 05/24/2023]
Abstract
We study the rigidity of two-dimensional site-diluted central force triangular networks under tension. We calculate the shear modulus micro directly and fit it with a power law of the form mu approximately (p-p*)(f), where p is the concentration of sites, p* its critical value, and f the critical exponent. We find that the critical behavior of mu is quite sensitive to tension. As the tension is increased there is at first a sharp drop in the values of both p* and f, followed by a slower decrease towards the values of the diluted Gaussian spring network (or random resistor network). We find that the size of the critical region is also sensitive to tension. The tension-free system has a narrower critical regime with the power law failing for p>0.8. In contrast, a small tension is sufficient to extend the power law to near p=1. The physical basis for these behaviors is discussed.
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Affiliation(s)
- Zicong Zhou
- Department of Physics and Institute of Life Sciences, Tamkang University, 151 Ying-chuan, Tamsui 25137, Taiwan, Republic of China.
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Discher BM, Bermudez H, Hammer DA, Discher DE, Won YY, Bates FS. Cross-linked Polymersome Membranes: Vesicles with Broadly Adjustable Properties. J Phys Chem B 2002. [DOI: 10.1021/jp011958z] [Citation(s) in RCA: 219] [Impact Index Per Article: 10.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Alava M, Duxbury P, Moukarzel C, Rieger H. Exact combinatorial algorithms: Ground states of disordered systems. ACTA ACUST UNITED AC 2001. [DOI: 10.1016/s1062-7901(01)80009-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/16/2023]
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de Menezes MA, Moukarzel CF. Directed rigidity and bootstrap percolation in 1+1 dimensions. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:5699-705. [PMID: 11970465 DOI: 10.1103/physreve.60.5699] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/06/1999] [Indexed: 04/18/2023]
Abstract
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.
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Affiliation(s)
- M A de Menezes
- Instituto de Física, Universidade Federal Fluminense, CEP 24210-340, Niteroi, RJ, Brazil.
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Arbabi S, Sahimi M. Absence of universality in percolation models of disordered elastic media with central forces. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/21/17/008] [Citation(s) in RCA: 20] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Sahimi M. Linear and nonlinear, scalar and vector transport processes in heterogeneous media: Fractals, percolation, and scaling laws. ACTA ACUST UNITED AC 1996. [DOI: 10.1016/s0923-0467(96)03103-x] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Moukarzel C, Duxbury PM. Stressed backbone and elasticity of random central-force systems. PHYSICAL REVIEW LETTERS 1995; 75:4055-4058. [PMID: 10059803 DOI: 10.1103/physrevlett.75.4055] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Boal DH. Rigidity and connectivity percolation in heterogeneous polymer-fluid networks. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 47:4604-4606. [PMID: 9960545 DOI: 10.1103/physreve.47.4604] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Arbabi S, Sahimi M. Mechanics of disordered solids. I. Percolation on elastic networks with central forces. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 47:695-702. [PMID: 10005543 DOI: 10.1103/physrevb.47.695] [Citation(s) in RCA: 92] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Plischke M, Fourcade B. Monte Carlo simulation of bond-diluted tethered membranes. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 43:2056-2058. [PMID: 9905249 DOI: 10.1103/physreva.43.2056] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Thorpe MF, Garboczi EJ. Elastic properties of central-force networks with bond-length mismatch. PHYSICAL REVIEW. B, CONDENSED MATTER 1990; 42:8405-8417. [PMID: 9995015 DOI: 10.1103/physrevb.42.8405] [Citation(s) in RCA: 46] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Yakubo K, Nakayama T. Fracton dynamics of percolating elastic networks: Energy spectrum and localized nature. PHYSICAL REVIEW. B, CONDENSED MATTER 1989; 40:517-523. [PMID: 9990943 DOI: 10.1103/physrevb.40.517] [Citation(s) in RCA: 43] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Garboczi EJ. Elastic softening versus amorphization in a simple model of ion-induced radiation damage. PHYSICAL REVIEW. B, CONDENSED MATTER 1989; 39:2472-2475. [PMID: 9948486 DOI: 10.1103/physrevb.39.2472] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Yan H, Day AR, Thorpe MF. Stability of networks under tension and pressure. PHYSICAL REVIEW. B, CONDENSED MATTER 1988; 38:6876-6880. [PMID: 9945368 DOI: 10.1103/physrevb.38.6876] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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Tang W, Thorpe MF. Percolation of elastic networks under tension. PHYSICAL REVIEW. B, CONDENSED MATTER 1988; 37:5539-5551. [PMID: 9943745 DOI: 10.1103/physrevb.37.5539] [Citation(s) in RCA: 60] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Garboczi EJ. Cauchy relations for central-force random networks. PHYSICAL REVIEW. B, CONDENSED MATTER 1987; 36:2115-2120. [PMID: 9943056 DOI: 10.1103/physrevb.36.2115] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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