Deformation of cross-linked semiflexible polymer networks

Bond breaking dynamics in semiflexible networks under load

We examine the bond-breaking dynamics of transiently cross-linked semiflexible networks using a single filament model in which that filament is peeled from an array of cross-linkers. We examine the effect of quenched disorder in the placement of the linkers along the filament and the effect of stochastic bond-breaking (assuming Bell model unbinding kinetics) on the dynamics of filament cross-linker dissociation and the statistics of ripping events. We find that bond forces decay exponentially away from the point of loading and that bond breaking proceeds sequentially down the linker array from the point of loading in a series of stochastic ripping events. We compare these theoretical predictions to the observed trajectories of large beads in a cross-linked microtubule network and identify the observed jumps of the bead with the linker rupture events predicted by the single filament model.

Entangled F-actin displays a unique crossover to microscale nonlinearity dominated by entanglement segment dynamics

We drive optically trapped microspheres through entangled F-actin at constant speeds and distances well beyond the linear regime, and measure the microscale force response of the entangled filaments during and following strain. Our results reveal a unique crossover to appreciable nonlinearity at a strain rate of [small gamma, Greek, dot above]c ≈ 3 s(-1) which corresponds remarkably well with the theoretical rate of relaxation of entanglement length deformations 1/τent. Above [small gamma, Greek, dot above]c, we observe stress stiffening which occurs over very short time scales comparable to the predicted timescale over which mesh size deformations relax. Stress softening then takes over, yielding to an effectively viscous regime over a timescale comparable to the entanglement length relaxation time, τent. The viscous regime displays shear thinning but with a less pronounced viscosity scaling with strain rate compared to flexible polymers. The relaxation of induced force on filaments following strain shows that the relative relaxation proceeds more quickly for increasing strain rates; and for rates greater than [small gamma, Greek, dot above]c, the relaxation displays a complex power-law dependence on time. Our collective results reveal that molecular-level nonlinear viscoelasticity is driven by non-classical dynamics of individual entanglement segments that are unique to semiflexible polymers.

Universality in the morphology and mechanics of coarsening amyloid fibril networks

Peptide hydrogels have important applications as biomaterials and in nanotechnology, but utilization often depends on their mechanical properties for which we currently have no predictive capability. Here we use a peptide model to simulate the formation of percolating amyloid fibril networks and couple these to the elastic network theory to determine their mechanical properties. We find that the time variation of network length scales can be collapsed onto master curves by using a time scaling function that depends on the peptide interaction anisotropy. The same scaling applies to network mechanics, revealing a nonmonotonic dependence of the shear modulus with time. Our structure-function relationship between the peptide building blocks, network morphology, and network mechanical properties can aid in the design of amyloid fibril networks with tailored mechanical properties.

Elasticity of 3D networks with rigid filaments and compliant crosslinks

Disordered filamentous networks with compliant crosslinks exhibit a low linear elastic shear modulus at small strains, but stiffen dramatically at high strains. Experiments have shown that the elastic modulus can increase by up to three orders of magnitude while the networks withstand relatively large stresses without rupturing. Here, we perform an analytical and numerical study on model networks in three dimensions. Our model consists of a collection of randomly oriented rigid filaments connected by flexible crosslinks that are modeled as wormlike chains. Due to zero probability of filament intersection in three dimensions, our model networks are by construction prestressed in terms of initial tension in the crosslinks. We demonstrate how the linear elastic modulus can be related to the prestress in these networks. Under the assumption of affine deformations in the limit of infinite crosslink density, we show analytically that the nonlinear elastic regime in 1- and 2-dimensional networks is characterized by power-law scaling of the elastic modulus with the stress. In contrast, 3-dimensional networks show an exponential dependence of the modulus on stress. Independent of dimensionality, if the crosslink density is finite, we show that the only persistent scaling exponent is that of the single wormlike chain. We further show that there is no qualitative change in the stiffening behavior of filamentous networks even if the filaments are bending-compliant. Consequently, unlike suggested in prior work, the model system studied here cannot provide an explanation for the experimentally observed linear scaling of the modulus with the stress in filamentous networks.

Mesoscale modelling of environmentally responsive hydrogels: emerging applications

We review recent advances in mesoscale computational modeling, focusing on dissipative particle dynamics, used to probe stimuli-sensitive behavior of hydrogels.

Determinants of fluidlike behavior and effective viscosity in cross-linked actin networks

The actin cortex has a well-documented ability to rapidly remodel and flow while maintaining long-range connectivity, but how this is achieved remains poorly understood. Here, we use computer simulations to explore how stress relaxation in cross-linked actin networks subjected to extensional stress depends on the interplay between network architecture and turnover. We characterize a regime in which a network response is nonaffine and stress relaxation is governed by the continuous dissipation of elastic energy via cyclic formation, elongation, and turnover of tension-bearing elements. Within this regime, for a wide range of network parameters, we observe a constant deformation (creep) rate that is linearly proportional to the rate of filament turnover, leading to a constant effective viscosity that is inversely proportional to turnover rate. Significantly, we observe a biphasic dependence of the creep rate on applied stress: below a critical stress threshold, the creep rate increases linearly with applied stress; above that threshold, the creep rate becomes independent of applied stress. We show that this biphasic stress dependence can be understood in terms of the nonlinear force-extension behavior of individual force-transmitting network elements. These results have important implications for understanding the origins and control of viscous flows both in the cortex of living cells and in other polymer networks.

Rheology of semiflexible bundle networks with transient linkers

We present a theoretical and computational analysis of the rheology of networks made up of bundles of semiflexible filaments bound by transient cross-linkers. Such systems are ubiquitous in the cytoskeleton and can be formed in vitro using filamentous actin and various cross-linkers. We find that their high-frequency rheology is characterized by a scaling behavior that is quite distinct from that of networks of the well-studied single semiflexible filaments. This regime can be understood theoretically in terms of a length-scale-dependent bending modulus for bundles. Next, we observe new dissipative dynamics associated with the shear-induced disruption of the network at intermediate frequencies. Finally, at low frequencies, we encounter a region of non-Newtonian rheology characterized by power-law scaling. This regime is dominated by bundle dissolution and large-scale rearrangements of the network driven by equilibrium thermal fluctuations.

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.

Advances in the mechanical modeling of filamentous actin and its cross-linked networks on multiple scales

The protein actin is a part of the cytoskeleton and, therefore, responsible for the mechanical properties of the cells. Starting with the single molecule up to the final structure, actin creates a hierarchical structure of several levels exhibiting a remarkable behavior. The hierarchy spans several length scales and limitations in computational power; therefore, there is a call for different mechanical modeling approaches for the different scales. On the molecular level, we may consider each atom in molecular dynamics simulations. Actin forms filaments by combining the molecules into a double helix. In a model, we replace molecular subdomains using coarse-graining methods, allowing the investigation of larger systems of several atoms. These models on the nanoscale inform continuum mechanical models of large filaments, which are based on worm-like chain models for polymers. Assemblies of actin filaments are connected with cross-linker proteins. Models with discrete filaments, so-called Mikado models, allow us to investigate the dependence of the properties of networks on the parameters of the constituents. Microstructurally motivated continuum models of the networks provide insights into larger systems containing cross-linked actin networks. Modeling of such systems helps to gain insight into the processes on such small scales. On the other hand, they call for verification and hence trigger the improvement of established experiments and the development of new methods.

Wrinkling of random and regular semiflexible polymer networks

We investigate wrinkling of two-dimensional random and triangular semiflexible polymer networks under shear. Both types of semiflexible networks exhibit wrinkling above a small critical shear angle, which scales with an exponent of the bending modulus between 1.9 and 2.0. Random networks exhibit hysteresis at the wrinkling threshold. Wrinkling lowers the total elastic energy by up to 20% and strongly affects the elastic properties of all semiflexible networks such as the crossover between bending and stretching dominated behavior. In random networks, we also find evidence for metastable wrinkled configurations. While the disordered microstructure of random networks affects the scaling behavior of wrinkle amplitudes, it has little effect on wrinkle wavelength. Therefore, wrinkles represent a robust, microstructure-independent assay of shear strain or elastic properties.

Predictive maps for stochastic nonaffine stiffening and damage in fibrous networks

The macroscopic responses of synthetic and natural filamentous networks are determined by a combination of microstructure and filament properties. Biofilament networks such as those of actin and fibrin have become vehicles for studying important concepts in mechanics such as rigidity percolation, linearity and nonlinearity, isotropy and anisotropy, affinity and nonaffinity, hardening and softening, bending and stretching transitions, etc. In this work, we consider generic fibrous network architectures to map out their mechanical responses over a wide range of filament properties. Using the finite element method, we perform two-dimensional simulations of discrete networks subjected to shear deformation. These simulations encompass stochastic effects arising from network topology (filament arrangement, orientation, and length distribution) and the thermally activated crosslink scission. We study the mechanics of these random networks up to a strain of 10%, including damage that is induced by crosslink scission. The response is nonlinear and the initial elastic modulus alone is not sufficient to give an understanding about the overall response. We show that the nonlinear elastic response of the network can be captured using a few parameters that depend on some well known length scales in network mechanics. For networks with filament density above the rigidity percolation threshold, by increasing filament density and bending stiffness, we observe a crossover from the bending dominated elastically compliant stiffening regime to a stretching dominated rigid nonstiffening regime. We show that in the bending dominated regime there are large deviations from the predictions of affine continuum theories. We also give a simple qualitative model for describing the contours of the incubation strain which marks the onset of damage in networks.

Convolution theory for dynamic systems: a bottom-up approach to the viscoelasticity of polymeric networks

Biological materials such as the cytoskeleton are characterized by remarkable viscoelastic properties and therefore represent the subject of numerous micro- and macrorheological experimental studies. By generalizing the previously introduced dynamic convolution theory (DCT) to two dimensions, we devise a bottom-up approach for the viscoelastic properties of extended, crosslinked semiflexible polymer networks. Brownian dynamics (BD) simulations serve to determine the dynamic linear self- and cross-response properties of isolated semiflexible polymers to externally applied forces and torques; these response functions are used as input to the DCT. For a given network topology, the frequency-dependent response of the network subject to a given external force/torque distribution is calculated via the DCT allowing to resolve both micro- and macrorheological properties of the networks. A mapping on continuum viscoelastic theory yields the corresponding viscoelastic bulk moduli. Special attention is drawn to the flexibility of crosslinkers, which couple angular degrees of freedom at the network nodes and which are found to sensitively affect the resulting rheological properties of the polymeric meshwork.

Alginate/PEO-PPO-PEO composite hydrogels with thermally-active plasticity

Stimuli-responsive hydrogels with high strength and toughness have received significant interest in recent years. Here, we report thermally active composite hydrogels comprising alginate and one of two poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) (PEO-PPO-PEO) triblock copolymers. Temperature-sensitive structural and mechanical changes are probed using calorimetry, neutron scattering, shear rheology, unconfined compression, and fracture. Below the lower gelation temperature, LGT, the mechanical properties are dominated by alginate. As the LGT is reached, the contribution of PEO-PPO-PEO to the mechanical properties is activated, resulting in order-of-magnitude increases in elastic modulus. Under compression, we show the evolution of plasticity for the composite hydrogels as the LGT is approached and surpassed, resulting in dramatic increases in fracture stress compared to neat alginate hydrogels. Plasticity was observed above the LGT and may be attributed to restructuring from the sliding of packed micelles and strain-hardening due to stress concentration on alginate cross-links and junction zones, ultimately leading to fracture.

Super-adhesive polymer-silica nanocomposite layers

Atomized spray plasma deposition (ASPD) using a precursor mixture of 2-hydroxyethyl methacrylate and methacryloyl-functionalized 15 nm silica nanoparticles leads to the formation of poly(2-hydroxyethyl methacrylate)-silica nanocomposite layers. The direct application of these coatings to overlapping glass-glass joints gives rise to excellent in situ adhesion reaching 84 MPa shear bond strength and 6 GPa shear modulus prior to the onset of adherent (bulk glass) failure. This significant enhancement in interfacial adhesion arises due to the silica nanoparticle surface methacryloyl groups enhancing cross-linking throughout the nanocomposite layer.

Fluctuation-stabilized marginal networks and anomalous entropic elasticity

We study the elastic properties of thermal networks of Hookean springs. In the purely mechanical limit, such systems are known to have a vanishing rigidity when their connectivity falls below a critical, isostatic value. In this work, we show that thermal networks exhibit a nonzero shear modulus G well below the isostatic point and that this modulus exhibits an anomalous, sublinear dependence on temperature T. At the isostatic point, G increases as the square root of T, while we find G∝Tα below the isostatic point, where α≃0.8. We show that this anomalous T dependence is entropic in origin.

Modeling of fibrin gels based on confocal microscopy and light-scattering data

Fibrin gels are biological networks that play a fundamental role in blood coagulation and other patho/physiological processes, such as thrombosis and cancer. Electron and confocal microscopies show a collection of fibers that are relatively monodisperse in diameter, not uniformly distributed, and connected at nodal points with a branching order of ∼3-4. Although in the confocal images the hydrated fibers appear to be quite straight (mass fractal dimension D(m) = 1), for the overall system 1<D(m)<2. Based on the confocal images, we developed a method to generate three-dimensional (3D) in silico gels made of cylindrical sticks of diameter d, density ρ, and average length <L>, joined at randomly distributed nodal points. The resulting 3D network strikingly resembles real fibrin gels and can be sketched as an assembly of densely packed fractal blobs, i.e., regions of size ξ, where the fiber concentration is higher than average. The blobs are placed at a distance ξ0 between their centers of mass so that they are overlapped by a factor η =ξ/ξ0 and have D(m) ∼1.2-1.6. The in silico gels' structure is quantitatively analyzed by its 3D spatial correlation function g(3D)(r) and corresponding power spectrum I(q) = FFT(3D[g3D(r)]), from which ρ, d, D(m), η, and ξ0 can be extracted. In particular, ξ0 provides an excellent estimate of the gel mesh size. The in silico gels' I(q) compares quite well with real gels' elastic light-scattering measurements. We then derived an analytical form factor for accurately fitting the scattering data, which allowed us to directly recover the gels' structural parameters.

Local mechanical response in semiflexible polymer networks subjected to an axisymmetric prestress

Analytical and numerical calculations are presented for the mechanical response of fiber networks in a state of axisymmetric prestress, in the limit where geometric nonlinearities such as fiber rotation are negligible. This allows us to focus on the anisotropy deriving purely from the nonlinear force-extension curves of individual fibers. The number of independent elastic coefficients for isotropic, axisymmetric, and fully anisotropic networks are enumerated before deriving expressions for the response to a locally applied force that can be tested against, e.g., microrheology experiments. Localized forces can generate anisotropy away from the point of application, so numerical integration of nonlinear continuum equations is employed to determine the stress field, and induced mechanical anisotropy, at points located directly behind and in front of a force monopole. Results are presented for the wormlike chain model in normalized forms, allowing them to be easily mapped to a range of systems. Finally, the relevance of these findings to naturally occurring systems and directions for future investigation are discussed.

Disorder-assisted melting and the glass transition in amorphous solids

The mechanical response of solids depends on temperature, because the way atoms and molecules respond collectively to deformation is affected at various levels by thermal motion. This is a fundamental problem of solid state science and plays a crucial role in materials science. In glasses, the vanishing of shear rigidity upon increasing temperature is the reverse process of the glass transition. It remains poorly understood due to the disorder leading to nontrivial (nonaffine) components in the atomic displacements. Our theory explains the basic mechanism of the melting transition of amorphous (disordered) solids in terms of the lattice energy lost to this nonaffine motion, compared to which thermal vibrations turn out to play only a negligible role. The theory is in good agreement with classic data on melting of amorphous polymers (for which no alternative theory can be found in the literature) and offers new opportunities in materials science.

Effective-medium theory of a filamentous triangular lattice

We present an effective-medium theory that includes bending as well as stretching forces, and we use it to calculate the mechanical response of a diluted filamentous triangular lattice. In this lattice, bonds are central-force springs, and there are bending forces between neighboring bonds on the same filament. We investigate the diluted lattice in which each bond is present with a probability p. We find a rigidity threshold p(b) which has the same value for all positive bending rigidity and a crossover characterizing bending, stretching, and bend-stretch coupled elastic regimes controlled by the central-force rigidity percolation point at p(CF)=/~2/3 of the lattice when fiber bending rigidity vanishes.

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.

Nonaffine displacements in crystalline solids in the harmonic limit

A systematic coarse graining of microscopic atomic displacements generates a local elastic deformation tensor D as well as a positive definite scalar χ measuring nonaffinity, i.e., the extent to which the displacements are not representable as affine deformations of a reference crystal. We perform an exact calculation of the statistics of χ and D and their spatial correlations for solids at low temperatures, within a harmonic approximation and in one and two dimensions. We obtain the joint distribution P(χ,D) and the two-point spatial correlation functions for χ and D. We show that nonaffine and affine deformations are coupled even in a harmonic solid, with a strength that depends on the size of the coarse-graining volume Ω and dimensionality. As a corollary to our work, we identify the field h(χ) conjugate to χ and show that this field may be tuned to produce a transition to a state where the ensemble average <χ> and the correlation length of χ diverge. Our work should be useful as a template for understanding nonaffine displacements in realistic systems with or without disorder and as a means for developing computational tools for studying the effects of nonaffine displacements in melting, plastic flow, and the glass transition.

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.

Impact of branching on the elasticity of actin networks

Actin filaments play a fundamental role in cell mechanics: assembled into networks by a large number of partners, they ensure cell integrity, deformability, and migration. Here we focus on the mechanics of the dense branched network found at the leading edge of a crawling cell. We develop a new technique based on the dipolar attraction between magnetic colloids to measure mechanical properties of branched actin gels assembled around the colloids. This technique allows us to probe a large number of gels and, through the study of different networks, to access fundamental relationships between their microscopic structure and their mechanical properties. We show that the architecture does regulate the elasticity of the network: increasing both capping and branching concentrations strongly stiffens the networks. These effects occur at protein concentrations that can be regulated by the cell. In addition, the dependence of the elastic modulus on the filaments' flexibility and on increasing internal stress has been studied. Our overall results point toward an elastic regime dominated by enthalpic rather than entropic deformations. This result strongly differs from the elasticity of diluted cross-linked actin networks and can be explained by the dense dendritic structure of lamellipodium-like networks.

Redundancy and cooperativity in the mechanics of compositely crosslinked filamentous networks

The cytoskeleton of living cells contains many types of crosslinkers. Some crosslinkers allow energy-free rotations between filaments and others do not. The mechanical interplay between these different crosslinkers is an open issue in cytoskeletal mechanics. Therefore, we develop a theoretical framework based on rigidity percolation to study a generic filamentous system containing both stretching and bond-bending forces to address this issue. The framework involves both analytical calculations via effective medium theory and numerical simulations on a percolating triangular lattice with very good agreement between both. We find that the introduction of angle-constraining crosslinkers to a semiflexible filamentous network with freely rotating crosslinks can cooperatively lower the onset of rigidity to the connectivity percolation threshold-a result argued for years but never before obtained via effective medium theory. This allows the system to ultimately attain rigidity at the lowest concentration of material possible. We further demonstrate that introducing angle-constraining crosslinks results in mechanical behaviour similar to just freely rotating crosslinked semflexible filaments, indicating redundancy and universality. Our results also impact upon collagen and fibrin networks in biological and bio-engineered tissues.

Computational modeling of axonal microtubule bundles under tension

Microtubule bundles cross-linked by tau protein serve a variety of neurological functions including maintaining mechanical integrity of the axon, promoting axonal growth, and facilitating cargo transport. It has been observed that axonal damage in traumatic brain injury leads to bundle disorientation, loss of axonal viability, and cognitive impairment. This study investigates the initial mechanical response of axonal microtubule bundles under uniaxial tension using a discrete bead-spring representation. Mechanisms of failure due to traumatic stretch loading and their impact on the mechanical response and stability are also characterized. This study indicates that cross-linked axonal microtubule bundles in tension display stiffening behavior similar to a power-law relationship from nonaffine network deformations. Stretching of cross-links and microtubule bending were the primary deformation modes at low stresses. Microtubule stretch was negligible up to tensile stresses of ∼1 MPa. Bundle failure occurred by failure of cross-links leading to pull-out of microtubules and loss of bundle integrity. This may explain the elongation, undulation, and delayed elasticity of axons following traumatic stretch loading. More extensively cross-linked bundles withstood higher tensile stresses before failing. The bundle mechanical behavior uncovered by these computational techniques should guide future experiments on stretch-injured axons.

Filament-length-controlled elasticity in 3D fiber networks

We present a model for disordered 3D fiber networks to study their linear and nonlinear elasticity. In contrast to previous 2D models, these 3D networks with binary crosslinks are underconstrained with respect to fiber stretching elasticity, suggesting that bending may dominate their response. We find that such networks exhibit a bending-dominated elastic regime controlled by fiber length, as well as a crossover to a stretch-dominated regime for long fibers. Finally, by extending the model to the nonlinear regime, we show that these networks become intrinsically nonlinear with a vanishing linear response regime in the limit of flexible or long filaments.

Nonlinear effective-medium theory of disordered spring networks

Disordered soft materials, such as fibrous networks in biological contexts, exhibit a nonlinear elastic response. We study such nonlinear behavior with a minimal model for networks on lattice geometries with simple Hookian elements with disordered spring constant. By developing a mean-field approach to calculate the differential elastic bulk modulus for the macroscopic network response of such networks under large isotropic deformations, we provide insight into the origins of the strain stiffening and softening behavior of these systems. We find that the nonlinear mechanics depends only weakly on the lattice geometry and is governed by the average network connectivity. In particular, the nonlinear response is controlled by the isostatic connectivity, which depends strongly on the applied strain. Our predictions for the strain dependence of the isostatic point as well as the strain-dependent differential bulk modulus agree well with numerical results in both two and three dimensions. In addition, by using a mapping between the disordered network and a regular network with random forces, we calculate the nonaffine fluctuations of the deformation field and compare them to the numerical results. Finally, we discuss the limitations and implications of the developed theory.

Mechanics of soft composites of rods in elastic gels

We report detailed theoretical investigations of the micromechanics and bulk elastic properties of composites consisting of randomly distributed stiff fibers embedded in an elastic matrix in two and three dimensions. Recent experiments [V. Pelletier, N. Gal, P. Fournier, and M. L. Kilfoil, Phys. Rev. Lett. 102, 188303 (2009)] have suggested that the inclusion of stiff microtubules in a softer, nearly incompressible biopolymer matrix can lead to emergent compressibility. This can be understood in terms of the enhancement of the compressibility of the composite relative to its shear compliance as a result of the addition of stiff rodlike inclusions. We show that the Poisson's ratio ν of such a composite evolves with increasing rod density toward a particular value, or fixed point, independent of the material properties of the matrix, as long as it has a finite initial compressibility. This fixed point is ν = 1/4 in three dimensions and ν = 1/3 in two dimensions. Our results suggest an important role for stiff filaments such as microtubules and stress fibers in cell mechanics. At the same time, our work has a wider elasticity context, with potential applications to composite elastic media with a wide separation of scales in stiffness of its constituents such as carbon nanotube-polymer composites, which have been shown to have highly tunable mechanics.

Rheology of heterotypic collagen networks

Collagen fibrils are the main structural element of connective tissues. In many tissues, these fibrils contain two fibrillar collagens (types I and V) in a ratio that changes during tissue development, regeneration, and various diseases. Here we investigate the influence of collagen composition on the structure and rheology of networks of purified collagen I and V, combining fluorescence and atomic force microscopy, turbidimetry, and rheometry. We demonstrate that the network stiffness strongly decreases with increasing collagen V content, even though the network structure does not substantially change. We compare the rheological data with theoretical models for rigid polymers and find that the elasticity is dominated by nonaffine deformations. There is no analytical theory describing this regime, hampering a quantitative interpretation of the influence of collagen V. Our findings are relevant for understanding molecular origins of tissue biomechanics and for guiding rational design of collagenous biomaterials for biomedical applications.

On the role of the filament length distribution in the mechanics of semiflexible networks

This paper explores the effects of filament length polydispersity on the mechanical properties of semiflexible crosslinked polymer networks. Extending previous studies on monodisperse networks, we compute numerically the response of crosslinked networks of elastic filaments of bimodal and exponential length distributions. These polydisperse networks are subject to the same affine to nonaffine (A/NA) transition observed previously for monodisperse networks, wherein the decreases in either crosslink density or bending stiffness lead to a shift from affine, stretching-dominated deformations to nonaffine, bending-dominated deformations. We find that the onset of this transition is generally more sensitive to changes in the density of longer filaments than shorter filaments, meaning that longer filaments have greater mechanical efficiency. Moreover, in polydisperse networks, mixtures of long and short filaments interact cooperatively to generally produce a nonaffine mechanical response closer to the affine prediction than comparable monodisperse networks of either long or short filaments. Accordingly, the mechanical affinity of polydisperse networks is dependent on the filament length composition. Overall, length polydispersity has the effect of sharpening and shifting the A/NA transition to lower network densities. We discuss the implications of these results on experimental observation of the A/NA transition, and on the design of advanced materials.

The influence of disorder on deformations in semiflexible networks

We present a model that assesses the different elastic responses of a semiflexible network, which either (i) is constrained to deform in an affine way or (ii) is permitted to thermally fluctuate and deviate from affine response. The thermal, non-affine response of the network is achieved using a Metropolis Monte Carlo algorithm with dynamic step size. We find that non-affine deformations soften the network dramatically at low strains and make the eventual nonlinear strain stiffening far more pronounced. We show that the effect of these non-affine deformations are very sensitive to the degree variation in the lengths of filaments connecting cross-links. Where there is high variation, non-affine deformations allow internal stresses to relax, giving rise to a smaller range of tensile forces in filaments and a dramatic reduction of network stiffness. This highlights that non-affine deformations are crucial in small strain response of stiff polymer networks.

Longitudinal response of confined semiflexible polymers

The longitudinal response of single semiflexible polymers to sudden changes in externally applied forces is known to be controlled by the propagation and relaxation of backbone tension. Under many experimental circumstances, realized, for example, in nanofluidic devices or in polymeric networks or solutions, these polymers are effectively confined in a channel- or tubelike geometry. By means of heuristic scaling laws and rigorous analytical theory, we analyze the tension dynamics of confined semiflexible polymers for various generic experimental setups. It turns out that in contrast to the well-known linear response, the influence of confinement on the nonlinear dynamics can largely be described as that of an effective prestress. We also study the free relaxation of an initially confined chain, finding a surprising superlinear ~t(9/8) growth law for the change in end-to-end distance at short times.

Coherent potential approximation of random nearly isostatic kagome lattice

The kagome lattice has coordination number 4, and it is mechanically isostatic when nearest-neighbor sites are connected by central-force springs. A lattice of N sites has O(√N) zero-frequency floppy modes that convert to finite-frequency anomalous modes when next-nearest-neighbor (NNN) springs are added. We use the coherent potential approximation to study the mode structure and mechanical properties of the kagome lattice in which NNN springs with spring constant κ are added with probability P=Δz/4, where Δz=z-4 and z is the average coordination number. The effective medium static NNN spring constant κ(m) scales as P(2) for P≪κ and as P for P≫κ, yielding a frequency scale ω*~Δz and a length scale l*~(Δz)(-1). To a very good approximation at small nonzero frequency, κ(m)(P,ω)/κ(m)(P,0) is a scaling function of ω/ω*. The Ioffe-Regel limit beyond which plane-wave states become ill-defined is reached at a frequency of order ω*.

Active force generation in cross-linked filament bundles without motor proteins

Cytoskeletal filaments often interact laterally through cross-linking proteins, contributing to passive cellular viscoelasticity and, perhaps surprisingly, active force generation. We present a theory, based on the formation and rupture of cross-linker bonds, that relates molecular properties of those interactions to the macroscale mechanics of filament bundles. Computing the force-velocity relation for such a bundle, we demonstrate significant contractile forces in the absence of molecular motors. This theory provides insight into cytokinesis, cytoskeletal mechanics, and stress-fiber contraction.

Insights into the potential functionality of single-chain force-induced conformational transitions in polymer networks: implications for polysaccharide signaling in the plant cell wall

The behavior of biopolymer networks comprised of clickable polysaccharide chains that can undergo force-induced conformational transitions was investigated during straining using a simulation technique. The simulation was carried out both using an affine deformation field and alternatively using Lees-Edwards boundary conditions as an example of a nonaffine case. In the affine situation the simulated stress-strain curves were found to be consistent with results obtained by evaluating the molecular force-extension curve at a single average extension and calculating the bulk modulus as an average over all possible orientations with respect to the deformation. While in all cases examined the macroscopic mechanical responses of networks of randomly oriented chains, consisting either of simple extensible wormlike chains or their clickable analogs, were found to be indistinguishable, the simulation additionally allowed the number of chains containing sugar rings in different conformational states to be monitored, and this was found to change significantly during straining. This supports the hypothesis that in networks of randomly oriented clickable polysaccharide chains, such conformational transitions could have biological significance as stress switches in signaling processes but that they are unlikely to affect the bulk rheological properties of tissue.

Affine-nonaffine transition in networks of nematically ordered semiflexible polymers

We study the mechanics of nematically ordered semiflexible networks showing that they, like isotropic networks, undergo an affine to nonaffine crossover controlled by the ratio of the filament length to the nonaffinity length. Deep in the nonaffine regime, however, these anisotropic networks exhibit a much more complex mechanical response characterized by a vanishing linear-response regime for highly ordered networks and a dependence of the shear modulus on shear direction at both small (linear) and finite (nonlinear) strains that is different from the affine prediction of orthotropic continuum linear elasticity. We show that these features can be understood in terms of a generalized floppy modes analysis of the nonaffine mechanics and a type of cooperative Euler buckling.

Poisson's ratio in composite elastic media with rigid rods

We calculate both the micromechanical response and bulk elastic constants of composites of rods embedded in elastic media. We find two fixed points for Poisson's ratio with respect to rod density: there is an unstable fixed point for Poisson's ratio =1/2 (an incompressible system) and a stable fixed point for Poisson's ratio =1/4 (a compressible system). We also derive approximate expressions for the elastic constants for arbitrary rod density, which agree with exact results for both low and high density. These results may help to explain recent experiments [Phys. Rev. Lett. 102, 188303 (2009)10.1103/PhysRevLett.102.188303] that reported compressibility for composites of microtubules in filamentous actin networks.

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.

Hierarchical models of rigidity percolation

We introduce models of generic rigidity percolation in two dimensions on hierarchical networks and solve them exactly by means of a renormalization transformation. We then study how the possibility for the network to self organize in order to avoid stressed bonds may change the phase diagram. In contrast to what happens on random graphs and in some recent numerical studies at zero temperature, we do not find a true intermediate phase separating the usual rigid and floppy ones.

Nonlocal fluctuation correlations in active gels

Many active materials and biological systems are driven far from equilibrium by embedded agents that spontaneously generate forces and distort the surrounding material. Probing and characterizing these athermal fluctuations are essential to understand the properties and behaviors of such systems. Here we present a mathematical procedure to estimate the local action of force-generating agents from the observed fluctuating displacement fields. The active agents are modeled as oriented force dipoles or isotropic compression foci, and the matrix on which they act is assumed to be either a compressible elastic continuum or a coupled network-solvent system. Correlations at a single point and between points separated by an arbitrary distance are obtained, giving a total of three independent fluctuation modes that can be tested with microrheology experiments. Since oriented dipoles and isotropic compression foci give different contributions to these fluctuation modes, ratiometric analysis allows us characterize the force generators. We also predict and experimentally find a high-frequency ballistic regime, arising from individual force-generating events in the form of the slow buildup of stress followed by rapid but finite decay. Finally, we provide a quantitative statistical model to estimate the mean filament tension from these athermal fluctuations, which leads to stiffening of active networks.