We study the mechanics of nematically ordered semiflexible networks showing that they, like isotropic networks, undergo an affine to non-affine cross-over controlled by the ratio of the filament length to the non-affinity length. Deep in the non-affine 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 generically more complex dependence of the shear modulus upon the direction of shear relative to the nematic director. We show that these features can be understood in terms of a generalized floppy modes analysis of the non-affine mechanics and a type of cooperative Euler buckling.
We image semi-flexible polymer networks under shear at the micrometer scale. By tracking embedded probe particles, we determine the local strain field, and directly measure its uniformity, or degree of affineness, on scales of 2-100 micron. The degree of nonaffine strain depends on polymer length and crosslink density, consistent with theoretical predictions. We also find a direct correspondence between the uniformity of the microscale strain and the nonlinear elasticity of the networks in the bulk.
We develop a free energy framework to describe the response of glasses to applied stress. Unlike crystals, for which the free energy increases quadratically with strain due to affine displacements, for glasses, the nonequilibrium free energy decreases due to complex interplay of non-affine displacements and dissipation. We measure this free energy directly in strained colloidal glasses, and use mean-field theory to relate it to affine and nonaffine displacements. Nonaffine displacements grow with applied shear due to shear-induced loss of structural connectivity. Our mean-field model allows fo…
We study dense and highly polydisperse emulsions at droplet volume fractions ϕ>= 0.65. We apply oscillatory shear and observe droplet motion using confocal microscopy. The presence of droplets with sizes several times the mean size dramatically changes the motion of smaller droplets. Both affine and nonaffine droplet motions are observed, with the more nonaffine motion exhibited by the smaller droplets which are pushed around by the larger droplets. Droplet motions are correlated over length scales from one to four times the mean droplet diameter, with larger length scales corresponding to hig…
Understanding the particle-scale transition from elastic deformation to plastic flow is central to making predictions about the bulk material properties and response of disordered materials. To address this issue, we perform experiments on flow-stabilized solids composed of micron-scale spheres within a microfluidic channel, in a regime where particle inertia is negligible. Each solid heap exists within a stress gradient imposed by the flow, and we track the positions of particles in response to single impulses of fluid-driven compression or decompression. We find that the resulting deformatio…
When an amorphous solid is deformed homogeneously, the response exhibits heterogeneous plastic instabilities with localized cooperative rearrangement of cluster of particles. The heterogeneous behavior plays an important role in deciding the mechanical properties of amorphous solids. In this paper, we employ computer simulation to study the characteristics and the spatial correlations of these clusters characterized by the non-affine displacements in amorphous solids under simple shear deformation in the athermal quasistatic limit. The clusters with large displacements are found to be homogene…
Materials characterized by spatially homogeneous elastic moduli undergo affine distortions when subjected to external stress at their boundaries, i.e., their displacements $\uv (\xv)$ from a uniform reference state grow linearly with position $\xv$, and their strains are spatially constant. Many materials, including all macroscopically isotropic amorphous ones, have elastic moduli that vary randomly with position, and they necessarily undergo nonaffine distortions in response to external stress. We study general aspects of nonaffine response and correlation using analytic calculations and nume…
Tissue remodelling in diverse developmental contexts require cell shape changes that have been associated with pulsation and flow of the actomyosin cytoskeleton. Here we describe the dynamics of the actomyosin cytoskeleton as a confined active elastomer embedded in the cytosol and subject to turnover of its components. Under affine deformations (homogeneous deformation over a spatially coarse-grained scale), the active elastomer exhibits spontaneous oscillations, propagating waves, contractile collapse and spatiotemporal chaos. The collective nonlinear dynamics shows nucleation, growth and coa…
Networks of semiflexible or stiff polymers such as most biopolymers are known to deform inhomogeneously when sheared. The effects of such non-affine deformation have been shown to be much stronger than for flexible polymers. To date, our understanding of non-affinity in such systems is limited to simulations or specific 2D models of athermal fibers. Here, we present an effective medium theory (EMT) for non-affine 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…
We investigate the viscoelastic properties of entangled networks of semiflexible polymers. At intermediate time scales the elastic response of these networks to shear deformation is described by the plateau modulus $G$. Different scaling laws with polymer concentration $c$ have been proposed based on the assumption that the deformation field is affine on all length scales. We develop a numerical approach that allows to calculate the modulus via free energy changes for both affine and non-affine deformations. The non-affine deformation field is obtained by a free energy minimization. Our findin…
In disordered materials under mechanical stress, the produced deformation can deviate from the affine one already in the elastic regime. The nonaffine contribution was observed and characterized in numerical simulations for various systems. However, low amplitude of nonaffinity and its local character makes the experimental study challenging. We present a novel method on the based on a phase compensation of the wave scattered from a dilated amorphous material using fine wavelength tuning of the optical probe beam. Using a glass frit as a sample, we provide complete reversibility of the materia…
The cytoskeleton of eukaryotic cells provides mechanical support and governs intracellular transport. These functions rely on the complex mechanical properties of networks of semiflexible protein filaments. Recent theoretical interest has focused on mesoscopic properties of such networks and especially on the effect of local, non-affine bending deformations on mechanics. Here, we study the impact of local network deformations on the scale-dependent mobility of probe particles in entangled networks of semiflexible actin filaments by high-bandwidth microrheology. We find that micron-sized partic…
Strain stiffening of protein networks is explored by means of a finite strain analysis of a two-dimensional network model of cross-linked semiflexible filaments. The results show that stiffening is caused by non-affine network rearrangements that govern a transition from a bending dominated response at small strains to a stretching dominated response at large strains. Thermally-induced filament undulations only have a minor effect; they merely postpone the transition.
We develop an effective medium approach to the mechanics of disordered, semiflexible polymer networks and study the response of such networks to uniform and nonuniform strain. We identify distinct elastic regimes in which the contributions of either filament bending or stretching to the macroscopic modulus vanish. We also show that our effective medium theory predicts a crossover between affine and non-affine strain, consistent with both prior numerical studies and scaling theory.
We present a revised theoretical study of the affine assumption applied to semiflexible networks. Drawing on simple models of semiflexible worm-like chains we derive an expression for the probability distribution of crosslink separations valid at all separations. This accounts for both entropic and mechanical filament stretching. From this we obtain the free energy density of such networks explicitly as a function of applied strain. We are therefore able to calculate the elastic moduli of such networks for any imposed strain or stress. We find that accounting for the distribution of cross-link…
Previous research on semiflexible polymers including cytoskeletal networks in cells has suggested the existence of distinct regimes of elastic response, in which the strain field is either uniform (affine) or non-uniform (non-affine) under external stress. Associated with these regimes, it has been further suggested that a new fundamental length scale emerges, which characterizes the scale for the crossover from non-affine to affine deformations. Here, we extend these studies by probing the response to localized forces and force dipoles. We show that the previously identified nonaffinity lengt…
A network of semiflexible biopolymers, known as the cytoskeleton, and molecular motors play fundamental mechanical roles in cellular activities. The cytoskeletal response to forces generated by molecular motors is profoundly linked to physiological processes. However, owing to the highly nonlinear mechanical properties, the cytoskeletal response on the microscopic level is largely elusive. The aim of this study is to investigate the microscopic mechanical response of semiflexible biopolymer networks by conducting microrheology (MR) experiments. Micrometer-sized colloidal particles, embedded in…
Reconstituted filamentous actin networks with myosin motor proteins form active gels, in which motor proteins generate forces that drive the network far from equilibrium. This motor activity can also strongly affect the network elasticity; experiments have shown a dramatic stiffening in in vitro networks with molecular motors. Here we study the effects of motor generated forces on the mechanics of simulated 2D networks of athermal stiff filaments. We show how heterogeneous internal motor stresses can lead to stiffening in networks that are governed by filament bending modes. The motors are mod…
Semiflexible biopolymer networks are commonly found in biological systems, from the cytoskeleton of cells to the extracellular matrix. Such networks often naturally occur as composites, in which various components interact to generate rich mechanical behaviours. In this work we examine the mechanics of composites formed when a single fibre inclusion is placed within discrete fibre networks of two distinct architectures. In particular, we computationally and theoretically investigate the mechanics of composites formed when an inclusion is introduced to Voronoi- and Mikado--type networks within …
While actin bundles are used by living cells for structural fortification, the microscopic origin of the elasticity of bundled networks is not understood. Here, we show that above a critical concentration of the actin binding protein fascin, a solution of actin filaments organizes into a pure network of bundles. While the elasticity of weakly crosslinked networks is dominated by the affine deformation of tubes, the network of bundles can be fully understood in terms of non-affine bending undulations.
Hydrogels of semiflexible biopolymers such as collagen have been shown to contract axially under shear strain, in contrast to the axial dilation observed for most elastic materials. Recent work has shown that this behavior can be understood in terms of the porous, two-component nature and consequent time-dependent compressibility of hydrogels. The apparent normal stress measured by a torsional rheometer reflects only the tensile contribution of the axial component $σ_{zz}$ on long (compressible) timescales, crossing over to the first normal stress difference, $N_1 = σ_{xx}-σ_{zz}$ at short (in…
The atomic theory of elasticity of amorphous solids, based on the nonaffine response formalism, is extended into the nonlinear stress-strain regime by coupling with the underlying irreversible many-body dynamics. The latter is implemented in compact analytical form using a qualitative method for the many-body Smoluchowski equation. The resulting nonlinear stress-strain (constitutive) relation is very simple, with few fitting parameters, yet contains all the microscopic physics. The theory is successfully tested against experimental data on metallic glasses, and it is able to reproduce the ubiq…
We present a theoretical framework for the linear and nonlinear visco-elastic properties of reversibly crosslinked networks of semiflexible polymers. In contrast to affine models where network strain couples to the polymer end-to-end distance, in our model strain rather serves to locally distort the network structure. This induces bending modes in the polymer filaments, the properties of wich are slaved to the surrounding network structure. Specifically, we investigate the frequency-dependent linear rheology, in particular in combination with crosslink binding/unbinding processes. We also deve…
Semiflexible polymers such as filamentous 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 predominantly affine and the filaments stretch an…
We show using Brownian dynamics simulations and theory how the shear relaxation modulus G(t) of dilute solutions of relatively stiff semiflexible polymers differs qualitatively from that of rigid rods. For chains shorter than their persistence length, G(t) exhibits three time regimes: At very early times, when the longitudinal deformation is affine, G(t) \sim t^{-3/4}. Over a broad intermediate regime, during which the chain length relaxes, G(t) \sim t^{-5/4}. At long times, $G(t) mimics that of rigid rods. A model of the polymer as an effectively extensible rod with a frequency dependent elas…
Bundles of polymer filaments are responsible for the rich and unique mechanical behaviors of many biomaterials, including cells and extracellular matrices. In fibrin biopolymers, whose nonlinear elastic properties are crucial for normal blood clotting, protofibrils self-assemble and bundle to form networks of semiflexible fibers. Here we show that the extraordinary strain-stiffening response of fibrin networks is a direct reflection of the hierarchical architecture of the fibrin fibers. We measure the rheology of networks of unbundled protofibrils and find excellent agreement with an affine mo…
Glasses acquire their solid-like properties by cooling from the supercooled liquid via a continuous transition known as the glass transition. Recent research on soft glasses indicates that besides temperature, another route to liquify glasses is by application of stress that forces relaxation and flow. Here we provide experimental evidence that the stress-induced onset of flow of glasses occurs via a sharp first order-like transition. Using simultaneous x-ray scattering during the oscillatory rheology of a colloidal glass, we identify a sharp symmetry change from anisotropic solid to isotropic…
Experiments have shown that elasticity of disordered filamentous networks with compliant crosslinks is very different from networks with rigid crosslinks. Here, we model and analyze filamentous networks as a collection of randomly oriented rigid filaments connected to each other by flexible crosslinks that are modeled as worm-like chains. For relatively large extensions we allow for enthalpic stretching of crosslinks' backbones. We show that for sufficiently high crosslink density, the network linear elastic response is affine on the scale of the filaments' length. The nonlinear regime can bec…
The actin cytoskeleton in living cells has many types of crosslinkers. The mechanical interplay between these different crosslinker types is an open issue in cytoskeletal mechanics. We develop a framework to study the cooperativity and redundancy in the mechanics of filamentous networks with two types of crosslinkers: crosslinkers that allow free rotations of filaments and crosslinkers that do not. The framework consists of numerical simulations and an effective medium theory on a percolating triangular lattice. We find that the introduction of angle-constraining crosslinkers significantly low…
