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Chen S, Markovich T, MacKintosh FC. Field Theory for Mechanical Criticality in Disordered Fiber Networks. PHYSICAL REVIEW LETTERS 2024; 133:028201. [PMID: 39073948 DOI: 10.1103/physrevlett.133.028201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2023] [Revised: 03/30/2024] [Accepted: 05/24/2024] [Indexed: 07/31/2024]
Abstract
Strain-controlled criticality governs the elasticity of jamming and fiber networks. While the upper critical dimension of jamming is believed to be d_{u}=2, non-mean-field exponents are observed in numerical studies of 2D and 3D fiber networks. The origins of this remains unclear. In this study we propose a minimal mean-field model for strain-controlled criticality of fiber networks. We then extend this to a phenomenological field theory, in which non-mean-field behavior emerges as a result of the disorder in the network structure. We predict that the upper critical dimension for such systems is d_{u}=4 using a Gaussian approximation. Moreover, we identify an order parameter for the phase transition, which has been lacking for fiber networks to date.
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Affiliation(s)
- Sihan Chen
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
- Center for Theoretical Biological Physics, Rice University, Houston, Texas 77005, USA
- Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, Illinois 60637, USA
- The James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
| | - Tomer Markovich
- Center for Theoretical Biological Physics, Rice University, Houston, Texas 77005, USA
- School of Mechanical Engineering, Tel Aviv University, Tel Aviv 69978, Israel
- Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv 69978, Israel
| | - Fred C MacKintosh
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
- Center for Theoretical Biological Physics, Rice University, Houston, Texas 77005, USA
- Department of Chemical and Biomolecular Engineering, Rice University, Houston, Texas 77005, USA
- Department of Chemistry, Rice University, Houston, Texas 77005, USA
- The Isaac Newton Institute for Mathematical Sciences, Cambridge University, Cambridge, United Kingdom
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2
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Arzash S, Gannavarapu A, MacKintosh FC. Mechanical criticality of fiber networks at a finite temperature. Phys Rev E 2023; 108:054403. [PMID: 38115508 DOI: 10.1103/physreve.108.054403] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2023] [Accepted: 10/05/2023] [Indexed: 12/21/2023]
Abstract
At zero temperature, spring networks with connectivity below Maxwell's isostatic threshold undergo a mechanical phase transition from a floppy state at small strains to a rigid state for applied shear strain above a critical strain threshold. Disordered networks in the floppy mechanical regime can be stabilized by entropic effects at finite temperature. We develop a scaling theory for this mechanical phase transition at finite temperature, yielding relationships between various scaling exponents. Using Monte Carlo simulations, we verify these scaling relations and identify anomalous entropic elasticity with sublinear T dependence in the linear elastic regime. While our results are consistent with prior studies of phase behavior near the isostatic point, the present work also makes predictions relevant to the broad class of disordered thermal semiflexible polymer networks for which the connectivity generally lies far below the isostatic threshold.
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Affiliation(s)
- Sadjad Arzash
- Department of Chemical and Biomolecular Engineering, Rice University, Houston, Texas 77005, USA
- Center for Theoretical Biological Physics, Rice University, Houston, Texas 77030, USA
| | - Anupama Gannavarapu
- Department of Chemical and Biomolecular Engineering, Rice University, Houston, Texas 77005, USA
- Center for Theoretical Biological Physics, Rice University, Houston, Texas 77030, USA
| | - Fred C MacKintosh
- Department of Chemical and Biomolecular Engineering, Rice University, Houston, Texas 77005, USA
- Center for Theoretical Biological Physics, Rice University, Houston, Texas 77030, USA
- Department of Chemistry and Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
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3
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Chen S, Markovich T, MacKintosh FC. Effective medium theory for mechanical phase transitions of fiber networks. SOFT MATTER 2023; 19:8124-8135. [PMID: 37846933 DOI: 10.1039/d3sm00810j] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/18/2023]
Abstract
Networks of stiff fibers govern the elasticity of biological structures such as the extracellular matrix of collagen. These networks are known to stiffen nonlinearly under shear or extensional strain. Recently, it has been shown that such stiffening is governed by a strain-controlled athermal but critical phase transition, from a floppy phase below the critical strain to a rigid phase above the critical strain. While this phase transition has been extensively studied numerically and experimentally, a complete analytical theory for this transition remains elusive. Here, we present an effective medium theory (EMT) for this mechanical phase transition of fiber networks. We extend a previous EMT appropriate for linear elasticity to incorporate nonlinear effects via an anharmonic Hamiltonian. The mean-field predictions of this theory, including the critical exponents, scaling relations and non-affine fluctuations qualitatively agree with previous experimental and numerical results.
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Affiliation(s)
- Sihan Chen
- Department of Physics and Astronomy, Rice University, Houston, TX 77005, USA.
- Center for Theoretical Biological Physics, Rice University, Houston, TX 77005, USA
| | - Tomer Markovich
- Center for Theoretical Biological Physics, Rice University, Houston, TX 77005, USA
- School of Mechanical Engineering, Tel Aviv University, Tel Aviv 69978, Israel
- Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv 69978, Israel
| | - Fred C MacKintosh
- Department of Physics and Astronomy, Rice University, Houston, TX 77005, USA.
- Center for Theoretical Biological Physics, Rice University, Houston, TX 77005, USA
- Department of Chemical and Biomolecular Engineering, Rice University, Houston, TX 77005, USA
- Department of Chemistry, Rice University, Houston, TX 77005, USA
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Chen S, Markovich T, MacKintosh FC. Nonaffine Deformation of Semiflexible Polymer and Fiber Networks. PHYSICAL REVIEW LETTERS 2023; 130:088101. [PMID: 36898114 DOI: 10.1103/physrevlett.130.088101] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/11/2022] [Accepted: 01/18/2023] [Indexed: 06/18/2023]
Abstract
Networks of semiflexible or stiff polymers such as most biopolymers are known to deform inhomogeneously when sheared. The effects of such nonaffine deformation have been shown to be much stronger than for flexible polymers. To date, our understanding of nonaffinity in such systems is limited to simulations or specific 2D models of athermal fibers. Here, we present an effective medium theory for nonaffine deformation of semiflexible polymer and fiber networks, which is general to both 2D and 3D and in both thermal and athermal limits. The predictions of this model are in good agreement with both prior computational and experimental results for linear elasticity. Moreover, the framework we introduce can be extended to address nonlinear elasticity and network dynamics.
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Affiliation(s)
- Sihan Chen
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
- Center for Theoretical Biological Physics, Rice University, Houston, Texas 77005, USA
| | - Tomer Markovich
- Center for Theoretical Biological Physics, Rice University, Houston, Texas 77005, USA
- School of Mechanical Engineering, Tel Aviv University, Tel Aviv 69978, Israel
- Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv 69978, Israel
| | - Fred C MacKintosh
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
- Center for Theoretical Biological Physics, Rice University, Houston, Texas 77005, USA
- Department of Chemical and Biomolecular Engineering, Rice University, Houston, Texas 77005, USA
- Department of Chemistry, Rice University, Houston, Texas 77005, USA
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Arzash S, Sharma A, MacKintosh FC. Mechanics of fiber networks under a bulk strain. Phys Rev E 2022; 106:L062403. [PMID: 36671162 DOI: 10.1103/physreve.106.l062403] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/02/2022] [Accepted: 12/01/2022] [Indexed: 06/17/2023]
Abstract
Biopolymer networks are common in biological systems from the cytoskeleton of individual cells to collagen in the extracellular matrix. The mechanics of these systems under applied strain can be explained in some cases by a phase transition from soft to rigid states. For collagen networks, it has been shown that this transition is critical in nature and it is predicted to exhibit diverging fluctuations near a critical strain that depends on the network's connectivity and structure. Whereas prior work focused mostly on shear deformation that is more accessible experimentally, here we study the mechanics of such networks under an applied bulk or isotropic extension. We confirm that the bulk modulus of subisostatic fiber networks exhibits similar critical behavior as a function of bulk strain. We find different nonmean-field exponents for bulk as opposed to shear. We also confirm a similar hyperscaling relation to what was previously found for shear.
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Affiliation(s)
- Sadjad Arzash
- Department of Chemical & Biomolecular Engineering, Rice University, Houston, Texas 77005, USA
- Center for Theoretical Biological Physics, Rice University, Houston, Texas 77030, USA
| | - Abhinav Sharma
- Leibniz-Institut für Polymerforschung Dresden, Institut Theorie der Polymere, 01069 Dresden, Germany
| | - Fred C MacKintosh
- Department of Chemical & Biomolecular Engineering, Rice University, Houston, Texas 77005, USA
- Center for Theoretical Biological Physics, Rice University, Houston, Texas 77030, USA
- Department of Chemistry, Rice University, Houston, Texas 77005, USA
- Department of Physics & Astronomy, Rice University, Houston, Texas 77005, USA
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Lee CT, Merkel M. Stiffening of under-constrained spring networks under isotropic strain. SOFT MATTER 2022; 18:5410-5425. [PMID: 35822259 DOI: 10.1039/d2sm00075j] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/15/2023]
Abstract
Disordered spring networks are a useful paradigm to examine macroscopic mechanical properties of amorphous materials. Here, we study the elastic behavior of under-constrained spring networks, i.e. networks with more degrees of freedom than springs. While such networks are usually floppy, they can be rigidified by applying external strain. Recently, an analytical formalism has been developed to predict the scaling behavior of the elastic network properties close to this rigidity transition. Here we numerically show that these predictions apply to many different classes of spring networks, including phantom triangular, Delaunay, Voronoi, and honeycomb networks. The analytical predictions further imply that the shear modulus G scales linearly with isotropic stress T close to the rigidity transition. However, this seems to be at odds with recent numerical studies suggesting an exponent between G and T that is smaller than one for some network classes. Using increased numerical precision and shear stabilization, we demonstrate here that close to the transition a linear scaling, G ∼ T, holds independent of the network class. Finally, we show that our results are not or only weakly affected by finite-size effects, depending on the network class.
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Affiliation(s)
- Cheng-Tai Lee
- CNRS, Centre de Physique Théorique (CPT, UMR 7332), Turing Center for Living Systems, Aix Marseille Univ, Université de Toulon, Marseille, France.
| | - Matthias Merkel
- CNRS, Centre de Physique Théorique (CPT, UMR 7332), Turing Center for Living Systems, Aix Marseille Univ, Université de Toulon, Marseille, France.
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Ruiz-Franco J, van Der Gucht J. Force Transmission in Disordered Fibre Networks. Front Cell Dev Biol 2022; 10:931776. [PMID: 35846368 PMCID: PMC9280074 DOI: 10.3389/fcell.2022.931776] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2022] [Accepted: 06/06/2022] [Indexed: 01/23/2023] Open
Abstract
Cells residing in living tissues apply forces to their immediate surroundings to promote the restructuration of the extracellular matrix fibres and to transmit mechanical signals to other cells. Here we use a minimalist model to study how these forces, applied locally by cell contraction, propagate through the fibrous network in the extracellular matrix. In particular, we characterize how the transmission of forces is influenced by the connectivity of the network and by the bending rigidity of the fibers. For highly connected fiber networks the stresses spread out isotropically around the cell over a distance that first increases with increasing contraction of the cell and then saturates at a characteristic length. For lower connectivity, however, the stress pattern is highly asymmetric and is characterised by force chains that can transmit stresses over very long distances. We hope that our analysis of force transmission in fibrous networks can provide a new avenue for future studies on how the mechanical feedback between the cell and the ECM is coupled with the microscopic environment around the cells.
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Tauber J, van der Gucht J, Dussi S. Stretchy and disordered: Toward understanding fracture in soft network materials via mesoscopic computer simulations. J Chem Phys 2022; 156:160901. [PMID: 35490006 DOI: 10.1063/5.0081316] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/20/2023] Open
Abstract
Soft network materials exist in numerous forms ranging from polymer networks, such as elastomers, to fiber networks, such as collagen. In addition, in colloidal gels, an underlying network structure can be identified, and several metamaterials and textiles can be considered network materials as well. Many of these materials share a highly disordered microstructure and can undergo large deformations before damage becomes visible at the macroscopic level. Despite their widespread presence, we still lack a clear picture of how the network structure controls the fracture processes of these soft materials. In this Perspective, we will focus on progress and open questions concerning fracture at the mesoscopic scale, in which the network architecture is clearly resolved, but neither the material-specific atomistic features nor the macroscopic sample geometries are considered. We will describe concepts regarding the network elastic response that have been established in recent years and turn out to be pre-requisites to understand the fracture response. We will mostly consider simulation studies, where the influence of specific network features on the material mechanics can be cleanly assessed. Rather than focusing on specific systems, we will discuss future challenges that should be addressed to gain new fundamental insights that would be relevant across several examples of soft network materials.
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Affiliation(s)
- Justin Tauber
- Physical Chemistry and Soft Matter, Wageningen University, Wageningen, The Netherlands
| | - Jasper van der Gucht
- Physical Chemistry and Soft Matter, Wageningen University, Wageningen, The Netherlands
| | - Simone Dussi
- Physical Chemistry and Soft Matter, Wageningen University, Wageningen, The Netherlands
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Damavandi OK, Hagh VF, Santangelo CD, Manning ML. Energetic rigidity. II. Applications in examples of biological and underconstrained materials. Phys Rev E 2022; 105:025004. [PMID: 35291184 DOI: 10.1103/physreve.105.025004] [Citation(s) in RCA: 9] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/08/2021] [Accepted: 01/24/2022] [Indexed: 06/14/2023]
Abstract
This is the second paper devoted to energetic rigidity, in which we apply our formalism to examples in two dimensions: Underconstrained random regular spring networks, vertex models, and jammed packings of soft particles. Spring networks and vertex models are both highly underconstrained, and first-order constraint counting does not predict their rigidity, but second-order rigidity does. In contrast, spherical jammed packings are overconstrained and thus first-order rigid, meaning that constraint counting is equivalent to energetic rigidity as long as prestresses in the system are sufficiently small. Aspherical jammed packings on the other hand have been shown to be jammed at hypostaticity, which we use to argue for a modified constraint counting for systems that are energetically rigid at quartic order.
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Affiliation(s)
- Ojan Khatib Damavandi
- Department of Physics and BioInspired Institute, Syracuse University, Syracuse, New York 13244, USA
| | - Varda F Hagh
- James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
| | - Christian D Santangelo
- Department of Physics and BioInspired Institute, Syracuse University, Syracuse, New York 13244, USA
| | - M Lisa Manning
- Department of Physics and BioInspired Institute, Syracuse University, Syracuse, New York 13244, USA
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Grill MJ, Kernes J, Slepukhin VM, Wall WA, Levine AJ. Directed force propagation in semiflexible networks. SOFT MATTER 2021; 17:10223-10241. [PMID: 33367438 DOI: 10.1039/d0sm01177k] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
We consider the propagation of tension along specific filaments of a semiflexible filament network in response to the application of a point force using a combination of numerical simulations and analytic theory. We find the distribution of force within the network is highly heterogeneous, with a small number of fibers supporting a significant fraction of the applied load over distances of multiple mesh sizes surrounding the point of force application. We suggest that these structures may be thought of as tensile force chains, whose structure we explore via simulation. We develop self-consistent calculations of the point-force response function and introduce a transfer matrix approach to explore the decay of tension (into bending) energy and the branching of tensile force chains in the network.
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Affiliation(s)
- Maximilian J Grill
- Institute for Computational Mechanics, Technical University of Munich, 85748 Garching, Germany
| | - Jonathan Kernes
- Department of Physics & Astronomy, University of California, Los Angeles, 90095, USA.
| | - Valentin M Slepukhin
- Department of Physics & Astronomy, University of California, Los Angeles, 90095, USA.
| | - Wolfgang A Wall
- Institute for Computational Mechanics, Technical University of Munich, 85748 Garching, Germany
| | - Alex J Levine
- Department of Physics & Astronomy, University of California, Los Angeles, 90095, USA.
- Department of Chemistry & Biochemistry, University of California, Los Angeles, 90095, USA
- Department of Computational Medicine, University of California, Los Angeles, 90095, USA
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Arzash S, Shivers JL, MacKintosh FC. Shear-induced phase transition and critical exponents in three-dimensional fiber networks. Phys Rev E 2021; 104:L022402. [PMID: 34525571 DOI: 10.1103/physreve.104.l022402] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/07/2021] [Accepted: 07/29/2021] [Indexed: 11/07/2022]
Abstract
When subject to applied strain, fiber networks exhibit nonlinear elastic stiffening. Recent theory and experiments have shown that this phenomenon is controlled by an underlying mechanical phase transition that is critical in nature. Growing simulation evidence points to non-mean-field behavior for this transition and a hyperscaling relation has been proposed to relate the corresponding critical exponents. Here, we report simulations on two distinct network structures in three dimensions. By performing a finite-size scaling analysis, we test hyperscaling and identify various critical exponents. From the apparent validity of hyperscaling, as well as the non-mean-field exponents we observe, our results suggest that the upper critical dimension for the strain-controlled phase transition is above three, in contrast to the jamming transition that represents another athermal, mechanical phase transition.
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Affiliation(s)
- Sadjad Arzash
- Department of Chemical & Biomolecular Engineering, Rice University, Houston, Texas 77005, USA.,Center for Theoretical Biological Physics, Rice University, Houston, Texas 77030, USA
| | - Jordan L Shivers
- Department of Chemical & Biomolecular Engineering, Rice University, Houston, Texas 77005, USA.,Center for Theoretical Biological Physics, Rice University, Houston, Texas 77030, USA
| | - Fred C MacKintosh
- Department of Chemical & Biomolecular Engineering, Rice University, Houston, Texas 77005, USA.,Center for Theoretical Biological Physics, Rice University, Houston, Texas 77030, USA.,Departments of Chemistry and Physics & Astronomy, Rice University, Houston, Texas 77005, USA
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