Rigidity and Dynamics of Random Spring Networks

Phononic band gap in random spring networks

We investigate the relation between topological and vibrational properties of networked materials by analyzing, both numerically and analytically, the properties of a random spring network model. We establish a pseudodispersion relation, which allows us to predict the existence of distinct transitions from extended to localized vibrational modes in this class of materials. Consequently, we propose an alternative method to control phonon and elastic wave propagation in disordered networks. In particular, the phonon band gap of our spring network model can be enhanced by either increasing its average degree or decreasing its assortativity coefficient. Applications to phonon band engineering and vibrational energy harvesting are briefly discussed.

Emergent power-law interactions in near-crystalline membranes

We derive exact results for the fluctuations in energy produced by microscopic disorder in near-crystalline athermal systems. Our formalism captures the heterogeneity in the elastic energy of polydisperse soft disks in energy-minimized configurations. We use this to predict the distribution of interaction energy between two defects in a disordered background. We show that this interaction energy displays a disorder-averaged power-law behavior 〈δE〉∼Δ^{-4} at large distances Δ between the defects. These interactions upon disorder average also display the sixfold symmetry of the underlying reference crystal. Additionally, we show that the fluctuations in the interaction energy encode the athermal correlations introduced by the disordered background. We verify our predictions with energy-minimized configurations of polydisperse soft disks in two dimensions.

Stiffest elastic networks

The rigidity of a network of elastic beams is closely related to its microstructure. We show both numerically and theoretically that there is a class of isotropic networks, which are stiffer than any other isotropic network of same density. The elastic moduli of these stiffest elastic networks are explicitly given. They constitute upper-bounds, which compete or improve the well-known Hashin–Shtrikman bounds. We provide a convenient set of criteria (necessary and sufficient conditions) to identify these networks and show that their displacement field under uniform loading conditions is affine down to the microscopic scale. Finally, examples of such networks with periodic arrangement are presented, in both two and three dimensions. In particular, we present an optimal and isotropic three-dimensional structure which, to our knowledge, is the first one to be presented as such.

Cross-Linked Fiber Network Embedded in Elastic Matrix

The mechanical behavior of a three-dimensional cross-linked fiber network embedded in matrix is studied in this work. The network is composed from linear elastic fibers which store energy only in the axial deformation mode, while the matrix is also isotropic and linear elastic. Such systems are encountered in a broad range of applications, from tissue to consumer products. As the matrix modulus increases, the network is constrained to deform more affinely. This leads to internal forces acting between the network and the matrix, which produce strong stress concentration at the network cross-links. This interaction increases the apparent modulus of the network and decreases the apparent modulus of the matrix. A model is developed to predict the effective modulus of the composite and its predictions are compared with numerical data for a variety of networks.

Stiffness transition in anisotropic fiber nets

We demonstrate the existence of a percolationlike stiffness transition in fiber networks with a bidisperse orientation distribution and with fiber densities clearly above the geometrical and the ordinary stiffness transition. The fibers are oriented parallel and perpendicular to a strain direction and they have a large fiber aspect ratio. The stiffness K of the fiber nets can be described by a scaling relation, K [proportionally] τ(α) g[(ε - ε(c))/τ(-β)], where τ is the fraction of fibers parallel to strain. g is a scaling function that is roughly described by a power law g(x) [proportionally ] x(γ) for stiffness above the transition and by a constant below the transition. The transition point is characterized by qualitative changes in the distribution of the elastic deformation energy of the fibers, the deformation mode of the fibers, the effective Poisson ratio of the nets, the distribution of elastic energy on fibers and cross links, and the ratio of elastic and viscous dissipation energy. This transition opens the possibility of extreme stiffness variations with minimal mesh manipulations in the vicinity of the transition (i.e., a stiffness gate). It is possible that this transition affects the mechanical behavior of the cytoskeleton in cells.

Model selection for athermal cross-linked fiber networks

Athermal random fiber networks are usually modeled by representing each fiber as a truss, a Euler-Bernoulli or a Timoshenko beam, and, in the case of cross-linked networks, each cross-link as a pinned, rotating, or welded joint. In this work we study the effect of these various modeling options on the dependence of the overall network stiffness on system parameters. We conclude that Timoshenko beams can be used for the entire range of density and beam stiffness parameters, while the Euler-Bernoulli model can be used only at relatively low network densities. In the high density-high bending stiffness range, strain energy is stored predominantly in the axial and shear deformation modes, while in the other extreme range of parameters, the energy is stored in the bending mode. The effect of the model size on the network stiffness is also discussed.

Energy distribution in disordered elastic networks

Disordered networks are found in many natural and artificial materials, from gels or cytoskeletal structures to metallic foams or bones. Here, the energy distribution in this type of networks is modeled, taking into account the orientation of the struts. A correlation between the orientation and the energy per unit volume is found and described as a function of the connectivity in the network and the relative bending stiffness of the struts. If one or both parameters have relatively large values, the struts aligned in the loading direction present the highest values of energy. On the contrary, if these have relatively small values, the highest values of energy can be reached in the struts oriented transversally. This result allows explaining in a simple way remodeling processes in biological materials, for example, the remodeling of trabecular bone and the reorganization in the cytoskeleton. Additionally, the correlation between the orientation, the affinity, and the bending-stretching ratio in the network is discussed.

Micromechanical model for elasticity of the cell cytoskeleton

Semiflexible polymer networks, such as cell cytoskeleton, differ significantly from their flexible counterparts in their deformation energy storage mechanism. As a result, the network elasticity is governed by both enthalpic and entropic variations. In addition, the enthalpic effect shows two distinct regimes of energy storage mechanism, the affine and nonaffine regimes. In the past, computation-based modeling on random networks, such as the Mikado model, was used to demonstrate the physical mechanism of mechanical deformation of semiflexible networks. These models are computationally intensive and hence are difficult to apply to studying whole cells. In this paper, we develop a micromechanical model to predict the average macroscopic elastic properties of a random, semiflexible, biopolymer network. The model employs a unit cell consisting of four semiflexible chains and four equivalent axial-bending springs. The proposed unit-cell-based micromechanical model represents a statistically average realization of the actual network and gives the average mechanical properties, such as the shear modulus. Comparisons between the model predictions and Mikado model results confirm that this micromechanical model captures the essential deformation physics revealed from previous studies on the actual network and is capable of predicting the transition between nonaffine and affine deformations. This model can be used to develop efficient continuum constitutive models of the cytoskeleton in the future.

"Bending to stretching" transition in disordered networks

From polymer gels to cytoskeletal structures, random networks of elastic material are commonly found in both materials science and biology. We present a three-dimensional micromechanical model of these networks and identify a "bending-to-stretching" transition. We characterize this transition in terms of concentration scaling laws, the stored elastic energy, and affinity measurements. Understanding the relationship between microscopic geometry and macroscopic mechanics will elucidate, for example, the mechanical properties of polymer gel networks or the role of semiflexible network mechanics in cells.

Unfolding cross-linkers as rheology regulators in F-actin networks

We report on the nonlinear mechanical properties of a statistically homogeneous, isotropic semiflexible network cross-linked by polymers containing numerous small unfolding domains, such as the ubiquitous F-actin cross-linker filamin. We show that the inclusion of such proteins has a dramatic effect on the large strain behavior of the network. Beyond a strain threshold, which depends on network density, the unfolding of protein domains leads to bulk shear softening. Past this critical strain, the network spontaneously organizes itself so that an appreciable fraction of the filamin cross-linkers are at the threshold of domain unfolding. We discuss via a simple mean-field model the cause of this network organization and suggest that it may be the source of power-law relaxation observed in in vitro and in intracellular microrheology experiments. We present data which fully justify our model for a simplified network architecture.

Floppy modes and nonaffine deformations in random fiber networks

We study the elasticity of random fiber networks. Starting from a microscopic picture of the nonaffine deformation fields, we calculate the macroscopic elastic moduli both in a scaling theory and a self-consistent effective medium theory. By relating nonaffinity to the low-energy excitations of the network ("floppy modes"), we achieve a detailed characterization of the nonaffine deformations present in fibrous networks.

Distinct regimes of elastic response and deformation modes of cross-linked cytoskeletal and semiflexible polymer networks

Semiflexible polymers such as filamentous actin (F-actin) play a vital role in the mechanical behavior of cells, yet the basic properties of cross-linked F-actin networks remain poorly understood. To address this issue, we have performed numerical studies of the linear response of homogeneous and isotropic two-dimensional networks subject to an applied strain at zero temperature. The elastic moduli are found to vanish for network densities at a rigidity percolation threshold. For higher densities, two regimes are observed: one in which the deformation is predominately affine and the filaments stretch and compress; and a second in which bending modes dominate. We identify a dimensionless scalar quantity, being a combination of the material length scales, that specifies to which regime a given network belongs. A scaling argument is presented that approximately agrees with this crossover variable. By a direct geometric measure, we also confirm that the degree of affinity under strain correlates with the distinct elastic regimes. We discuss the implications of our findings and suggest possible directions for future investigations.

Deformation of cross-linked semiflexible polymer networks

Networks of filamentous proteins play a crucial role in cell mechanics. These cytoskeletal networks, together with various cross-linking and other associated proteins largely determine the (visco)elastic response of cells. In this Letter we study a model system of cross-linked, stiff filaments in order to explore the connection between the microstructure under strain and the macroscopic response of cytoskeletal networks. We find two distinct regimes as a function primarily of cross-link density and filament rigidity: one characterized by affine deformation and one by nonaffine deformation. We characterize the crossover between these two.

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.

Dynamic rigidity transition

An inflated closed loop (or membrane) is used to demonstrate a dynamic rigidity transition that occurs when impact energy is added to the loop in static equilibrium at zero temperature. The only relevant parameter in this transition is the ratio of the energy needed to collapse the loop and the impact energy. When this ratio is below a threshold value close to unity, the loop collapses into a high-entropy floppy state, and it does not return to the rigid state unless the impact energy can escape. The internal oscillations are in the floppy state dominated by 1/f(2) noise. When the ratio is above the threshold, the loop does not collapse, and the internal oscillations resulting from the impact are dominated by the eigenfrequencies of the stretched membrane. In this state, the loop can bounce for a long time. It is still an open question whether bouncing will eventually vanish or whether a stationary bouncing state will be reached. The dynamic transition between the floppy and the rigid state is discontinuous.

Rigidity of random networks of stiff fibers in the low-density limit

Rigidity percolation is analyzed in two-dimensional random networks of stiff fibers. As fibers are randomly added to the system there exists a density threshold q=q(min) above which a rigid stress-bearing percolation cluster appears. This threshold is found to be above the connectivity percolation threshold q=q(c) such that q(min)=(1.1698+/-0.0004)q(c). The transition is found to be continuous, and in the universality class of the two-dimensional central-force rigidity percolation on lattices. At percolation threshold the rigid backbone of the percolating cluster was found to break into rigid clusters, whose number diverges in the limit of infinite system size, when a critical bond is removed. The scaling with system size of the average size of these clusters was found to give a new scaling exponent delta=1.61+/-0.04.

Rigidity transition in two-dimensional random fiber networks

Rigidity percolation is analyzed in two-dimensional random fibrous networks. The model consists of central forces between the adjacent crossing points of the fibers. Two strategies are used to incorporate rigidity: adding extra constraints between second-nearest crossing points with a probability p(sn), and "welding" individual crossing points by adding there four additional constraints with a probability p(weld), and thus fixing the angles between the fibers. These additional constraints will make the model rigid at a critical probability p(sn)=p(sn)(c) and p(weld)=p(weld)(c), respectively. Accurate estimates are given for the transition thresholds and for some of the associated critical exponents. The transition is found in both cases to be in the same universality class as that of the two-dimensional central-force rigidity percolation in diluted lattices.

Elasticity of Poissonian fiber networks

An effective-medium model is introduced for the elasticity of two-dimensional random fiber networks. These networks are commonly used as basic models of heterogeneous fibrous structures such as paper. Using the exact Poissonian statistics to describe the microscopic geometry of the network, the tensile modulus can be expressed by a single-parameter function. This parameter depends on the network density and fiber dimensions, which relate the macroscopic modulus to the relative importance of axial and bending deformations of the fibers. The model agrees well with simulation results and experimental findings. We also discuss the possible generalizations of the model.