We study the role of connectivity on the linear and nonlinear elastic behavior of amorphous systems using a two-dimensional random network of harmonic springs as a model system. A natural characterization of these systems arises in terms of the network coordination relative to that of an isostatic network $δz$; a floppy network has $δz<0$, while a stiff network has $δz>0$. Under the influence of an externally applied load we observe that the response of both floppy and rigid network are controlled by the same critical point, corresponding to the onset of rigidity. We use numerical simulations to compute the exponents which characterize the shear modulus, the amplitude of non-affine displacements, and the network stiffening as a function of $δz$, derive these theoretically and make predictions for the mechanical response of glasses and fibrous networks.
Floppy Networks (FNs) provide valuable insight into the origin of anomalous mechanical and thermal properties in soft matter systems, from polymers, rubber, and biomolecules to glasses and granular materials. Here, we use the very same FN concept to construct a quantitative microscopic theory of empty perovskites, a family of crystals with ReO$_3$ structure, which exhibit a number of unusual properties. One remarkable example is ScF$_3$, which shows a near-zero-temperature structural instability and large negative thermal expansion (NTE). We trace these effects to an FN-like crystalline archit…
Spatial segregation in the wall normal direction is investigated in suspensions containing a binary mixture of Neo-Hookean capsules subjected to pressure driven flow in a planar slit. The two components of the binary mixture have unequal membrane rigidities. The problem is studied numerically using an accelerated implementation of the boundary integral method. The effect of a variety of parameters was investigated, including the capillary number, rigidity ratio between the two species, volume fraction, confinement ratio, and the number fraction of the more floppy particle $X_f$ in the mixture.…
Inspired by the ubiquity of composite filamentous networks in nature we investigate models of biopolymer networks that consist of interconnected floppy and stiff filaments. Numerical simulations carried out in three dimensions allow us to explore the microscopic partitioning of stresses and strains between the stiff and floppy fractions c_s and c_f, and reveal a non-trivial relationship between the mechanical behavior and the relative fraction of stiff polymer: when there are few stiff polymers, non-percolated stiff ``inclusions`` are protected from large deformations by an encompassing floppy…
Gels of semi-flexible polymers, network glasses made of low valence elements, softly compressed ellipses and dense suspensions under flow are examples of floppy materials. These systems present collective motions with almost no restoring force. We study theoretically and numerically the frequency-dependence of the response of these materials, and the length scales that characterize their elasticity. We show that isotropic floppy elastic networks present a phonon gap for frequencies smaller than a frequency $ω^*$ governed by coordination, and that the elastic response is localized on a length s…
We study the elasticity of random fiber networks. Starting from a microscopic picture of the non-affine deformation fields we calculate the macroscopic elastic moduli both in a scaling theory and a self-consistent effective medium theory. By relating non-affinity to the low-energy excitations of the network (``floppy-modes'') we achieve a detailed characterization of the non-affine deformations present in fibrous networks.
Using a mean field approach and simulation, we study the non-linear mechanical response of the vertex model (VM) of biological tissue under compression and dilation. The VM is known to exhibit a transition between rigid and fluid-like, or floppy, states driven by geometric incompatibility. Target perimeter and area set a target shape which may not be geometrically achievable, thereby engendering frustration. Previously, an asymmetry in the linear elastic response was identified at the rigidity transition between compression and dilation. Here we show and characterize how the asymmetry extends …
We study the elasticity of a two-dimensional random network of rigid rods (``Mikado model''). The essential features incorporated into the model are the anisotropic elasticity of the rods and the random geometry of the network. We show that there are three distinct scaling regimes, characterized by two distinct length scales on the elastic backbone. In addition to a critical rigidiy percolation region and a homogeneously elastic regime we find a novel intermediate scaling regime, where elasticity is dominated by bending deformations.
Motivated by recent experiments showing nonlinear elasticity of in vitro networks of the biopolymer actin cross-linked with filamin, we present an effective medium theory of flexibly cross-linked stiff polymer networks. We model such networks by randomly oriented elastic rods connected by flexible connectors to a surrounding elastic continuum, which self-consistently represents the behavior of the rest of the network. This model yields a crossover from a linear elastic regime to a highly nonlinear elastic regime that stiffens in a way quantitatively consistent with experiment.
Model lattices consisting of balls connected by central-force springs provide much of our understanding of mechanical response and phonon structure of real materials. Their stability depends critically on their coordination number $z$. $d$-dimensional lattices with $z=2d$ are at the threshold of mechanical stability and are isostatic. Lattices with $z<2d$ exhibit zero-frequency "floppy" modes that provide avenues for lattice collapse. The physics of systems as diverse as architectural structures, network glasses, randomly packed spheres, and biopolymer networks is strongly influenced by a near…
We present a model for disordered 3D fiber networks to study their linear and nonlinear elasticity over a wide range of network densities and fiber lengths. In contrast to previous 2D models, these 3D networks with binary cross-links are under-constrained with respect to fiber stretching elasticity, suggesting that bending may dominate their response. We find that such networks exhibit a fiber length-controlled bending regime and a crossover to a stretch-dominated regime for lengths beyond a characteristic scale that depends on the fiber's elastic properties. Finally, by extending the model to…
We present a statistical mechanical study of stiff polymers, motivated by experiments on actin filaments and the considerable current interest in polymer networks. We obtain simple, approximate analytical forms for the force-extension relations and compare these with numerical treatments. We note the important role of boundary conditions in determining force-extension relations. The theoretical predictions presented here can be tested against single molecule experiments on neurofilaments and cytoskeletal filaments like actin and microtubules. Our work is motivated by the buckling of the cytosk…
We present a theory for the elasticity of cross-linked stiff polymer networks. Stiff polymers, unlike their flexible counterparts, are highly anisotropic elastic objects. Similar to mechanical beams stiff polymers easily deform in bending, while they are much stiffer with respect to tensile forces (``stretching''). Unlike in previous approaches, where network elasticity is derived from the stretching mode, our theory properly accounts for the soft bending response. A self-consistent effective medium approach is used to calculate the macroscopic elastic moduli starting from a microscopic charac…
We examine how the distribution of contour lengths and the high-stretch stiffening of individual chain segments affect the macroscopic shear modulus of flexible polymer gels, using a 2D numerical model, in which polymer segments form a triangular network and disorder is introduced by varying their contour lengths. We show that in the relevant parameter range: (i) the non-affine contribution to the shear modulus is negligible, i.e. the Born approximation is satisfactory; (ii) the shear modulus is dominated by the contribution originating from equilibrium chain tensions. Moreover, mechanical equ…
Structural Relaxation and Anisotropic Elasticity of Ordered Block Copolymer Melts
Department of Chemistry, Texas A\&M University, College Station, TX 77843, USA；Department of Materials Science；Engineering, Texas A\&M University, College Station, TX 77843, USA
Filamentous bio-materials such as fibrin or collagen networks exhibit an enormous stiffening of their elastic moduli upon large deformations. This pronounced nonlinear behavior stems from a significant separation between the stiffnesses scales associated with bending vs. stretching the material's constituent elements. Here we study a simple model of such materials - floppy networks of hinged rigid bars embedded in an elastic matrix - in which the effective ratio of bending to stretching stiffnesses vanishes identically. We introduce a theoretical framework and build upon it to construct a nume…
In this paper we focus on the mechanical properties of oligomeric glasses (waxes), employing a microscopic model that provides, via numerical simulations, information about the shear modulus of such materials, the failure mechanism via plastic instabilities and about the geometric responses of the oligomers themselves to a mechanical load. We present a microscopic theory that explains the numerically observed phenomena, including an exact theory of the shear modulus and of the plastic instabilities, both local and system spanning. In addition we present a model to explain the geometric changes…
We develop a continuum theory for equilibrium elasticity of a network of crosslinked semiflexible filaments, spanning the full range between flexible entropy-driven chains to stiff athermal rods. We choose the 3-chain constitutive model over several plausible candidates, and derive analytical expressions for the elastic energy at arbitrary strain, with the corresponding stress-strain relationship. The theory fits well to a wide range of experimental data on simple shear in different filament networks, quantitatively matching the differential shear modulus variation with stress, with only two a…
Unlike most synthetic materials, biological materials often stiffen as they are deformed. This nonlinear elastic response, critical for the physiological function of some tissues, has been documented since at least the 19th century, but the molecular structure and the design principles responsible for it are unknown. Current models for this response require geometrically complex ordered structures unique to each material. In this Article we show that a much simpler molecular theory accounts for strain stiffening in a wide range of molecularly distinct biopolymer gels formed from purified cytos…
We present theoretical and experimental studies of the elastic response of fibrous networks subjected to uniaxial strain. Uniaxial compression or extension is applied to extracellular networks of fibrin and collagen using a shear rheometer with free water in/outflow. Both axial stress and the network shear modulus are measured. Prior work [van Oosten et al., Scientific Reports, 2015, 6, 19270] has shown softening/stiffening of these networks under compression/extension, together with a nonlinear response to shear, but the origin of such behaviour remains poorly understood. Here, we study how u…
We study the elasticity of random stiff fiber networks. The elastic response of the fibers is characterized by a central force stretching stiffness as well as a bending stiffness that acts transverse to the fiber contour. Previous studies have shown that this model displays an anomalous elastic regime where the stretching mode is fully frozen out and the elastic energy is completely dominated by the bending mode. We demonstrate by simulations and scaling arguments that, in contrast to the bending dominated \emph{elastic energy}, the equally important \emph{elastic forces} are to a large extent…
The effective elasticity of highly charged stiff polyelectrolytes is studied in the presence of counterions, with and without added salt. The rigid polymer conformations may become unstable due to an effective attraction induced by counterion density fluctuations. Instabilities at the longest, or intermediate length scales may signal collapse to globule, or necklace states, respectively. In the presence of added-salt, a generalized electrostatic persistence length is obtained, which has a nontrivial dependence on the Debye screening length.
We perform simulations to numerically study the writhe distribution of a stiff polymer. We compare with analytic results of Bouchiat and Mezard (PRL 80 1556- (1998); cond-mat/9706050).
One problem of non-crystalline condensed matter (soft matter) is creating the right equilibrium between elasticity and viscosity, referred to as viscoelasticity. Manifestations of that can be found in everyday live, where the viscoelasticity in a tire needs to be balanced so it is still flexible and can dissipate shock-energy, yet hard enough for energy-saving operation. Similarly, the cartilage in joints needs to absorb shocks while operating at low- level friction with high elasticity. Two such examples with a biological applicability are stiff membranes, which allow for the sliding of joint…
We investigate the emergence of isotropic linear elasticity in amorphous and polycrystalline solids, via extensive numerical simulations. We show that the elastic properties are correlated over a finite length scale $ξ_E$, so that the central limit theorem dictates the emergence of continuum linear isotropic elasticity on increasing the specimen size. The stiffness matrix of systems of finite size $L > ξ_E$ is obtained adding to that predicted by linear isotropic elasticity a random one of spectral norm $(L/ξ_E)^{-3/2}$, in three spatial dimensions. We further demonstrate that the elastic leng…
We study the elasticity of fibrous materials composed of generalized stiff polymers. It is shown that in contrast to cellular foam-like structures affine strain fields are generically unstable. Instead, a subtle interplay between the architecture of the network and the elastic properties of its building blocks leads to intriguing mechanical properties with intermediate asymptotic scaling regimes. We present exhaustive numerical studies based on a finite element method complemented by scaling arguments.
Nonlinear elastic responses of short and stiff polyelectrolytes are investigated by dynamic simulations on a single molecule level. When a polyelectrolyte condensate undergoes a mechanical unfolding, two types of force-extension curves, i.e., a force plateau and a stick-release pattern, are observed depending on the strength of the electrostatic interaction. We provide a physical interpretation of such force-extension behavior in terms of intramolecular structures of the condensates. We also describe a charge distribution of condensed counterions onto a highly stretched polyelectrolyte, which …
The two approaches to analyzing the large strain behavior of rubbery networks are phenomenologically, using strain energy functions drawn from continuum mechanics, and molecular models, which apply statistical mechanics to compute the effect of chain orientation on the entropy. The early rubber elasticity models ignored intermolecular interactions, whereas later developments ("constraint models") included the effect of entanglements or steric constraints on the mechanical stress. These constitutive equations for rubber elasticity are compared to experimental results, and the connection of netw…
Networks with only central force interactions are floppy when their average connectivity is below an isostatic threshold. Although such networks are mechanically unstable, they can become rigid when strained. It was recently shown that the transition from floppy to rigid states as a function of simple shear strain is continuous, with hallmark signatures of criticality (Nat. Phys. 12, 584 (2016)). The nonlinear mechanical response of collagen networks was shown to be quantitatively described within the framework of such mechanical critical phenomenon. Here, we provide a more quantitative charac…
We use optical tweezers microrheology and fluorescence microscopy to apply nonlinear microscale strains to entangled and crosslinked actin networks, and measure the resulting stress and actin filament deformations. We couple nonlinear stress response and subsequent relaxation to the velocity profiles of individual fluorescent-labeled actin segments at varying times throughout the strain and varying distances from the strain path to determine the underlying molecular dynamics that give rise to the debated nonlinear response and stress propagation of crosslinked and entangled actin networks at t…
Recent experiments have demonstrated that the nonlinear elasticity of in vitro networks of the biopolymer actin is dramatically altered in the presence of a flexible cross-linker such as the abundant cytoskeletal protein filamin. The basic principles of such networks remain poorly understood. Here we describe an effective medium theory of flexibly cross-linked stiff polymer networks. We argue that the response of the cross-links can be fully attributed to entropic stiffening, while softening due to domain unfolding can be ignored. The network is modeled as a collection of randomly oriented rod…
A general approach is presented for understanding the stress response function in anisotropic granular layers in two dimensions. The formalism accommodates both classical anisotropic elasticity theory and linear theories of anisotropic directed force chain networks. Perhaps surprisingly, two-peak response functions can occur even for classical, anisotropic elastic materials, such as triangular networks of springs with different stiffnesses. In such cases, the peak widths grow linearly with the height of the layer, contrary to the diffusive spreading found in `stress-only' hyperbolic models. In…
A new method is used to measure the stress and elastic constants of purely entropic phantom networks, in which a fraction $p$ of neighbors are tethered by inextensible bonds. We find that close to the percolation threshold $p_c$ the shear modulus behaves as $(p-p_c)^f$, where the exponent $f\approx 1.35$ in two dimensions, and $f\approx 1.95$ in three dimensions, close to the corresponding values of the conductivity exponent in random resistor networks. The components of the stiffness tensor (elastic constants) of the spanning cluster follow a power law $\sim(p-p_c)^g$, with an exponent $g\app…
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 i…
Biological processes, from morphogenesis to tumor invasion, spontaneously generate shear stresses inside living tissue. The mechanisms that govern the transmission of mechanical forces in epithelia and the collective response of the tissue to bulk shear deformations remain, however, poorly understood. Using a minimal cell-based computational model, we investigate the constitutive relation of confluent tissues under simple shear deformation. We show that an initially undeformed fluidlike tissue acquires finite rigidity above a critical applied strain. This is akin to the shear-driven rigidity o…
