Nonlinear elasticity of semiflexible filament networks

From Seaweeds to Hydrogels: Recent Progress in Kappa-2 Carrageenans

Hybrid carrageenans, also called kappa-2 (K2) or weak kappa, are a class of sulfated polysaccharides with thermo-reversible gelling properties in water and are extracted from a specific family of red seaweeds. K2 are known in the industry for their texturizing properties which are intermediate between those of kappa-carrageenans (K) and iota-carrageenans (I). As such, K2 are gaining industrial interest, as they can replace blends of K and I (K + I) in some niche applications. Over the last decade or so, some progress has been made in unravelling K2's chemical structure. The understanding of K2 gel's structure-rheological properties' relationships has also improved. Such recent progress is reported here, reviewing the literature on gelling K2 published since the last review on the topic. The focus is on the seaweeds used for extracting K2, their block copolymer chemical structures, and how these impact on the gel's formation and rheological properties. The outcome of this review is that additional rheological and structural studies of K2 hydrogels are needed, in particular to understand their viscoelastic behavior under large deformation and to unravel the differences between the texturizing properties of K2 and K + I.

Synthetic strain-stiffening hydrogels towards mechanical adaptability

Living organisms are made of wet, soft tissues. However, there is only one candidate to simultaneously replicate the mechanical and composition features of load-bearing tissues, that is, strain-stiffening hydrogels. The conventional mechanical match design principle is mostly limited to stiffness matching. However, this strategy cannot sufficiently and necessarily lead to mechanical matching over the whole physiologic deformation period for tissues and damages the tissues over time. In this review, we aim to provide a comprehensive summary of the reported synthetic strain-stiffening hydrogels and particularly focus on the relationship between their structure and performance. Initially, we present a brief introduction on the significance of strain-stiffening hydrogels in mimicking the mechanics of tissues, and then we discuss the qualitative evaluation of the strain-stiffening behaviors to guide the design of materials towards mimicking soft tissue. After distinguishing the mechanical testing methods, we focus on the methods for the preparation of typical strain-stiffening hydrogels based on categories, such as network without strand entanglement, semiflexible network, and anisotropic networks. Subsequently, we discuss the structural evolution of strain-stiffening hydrogels. We hope that this review will serve as an updated introduction and reference for researchers who are interested in exploring strain-stiffening hydrogels as tissue-mimics for addressing the societal needs at various frontiers.

Hyperelastic continuum models for isotropic athermal fibrous networks

Many biological materials contain fibrous protein networks as their main structural components. Understanding the mechanical properties of such networks is important for creating biomimicking materials for cell and tissue engineering, and for developing novel tools for detecting and diagnosing disease. In this work, we develop continuum models for isotropic, athermal fibrous networks by combining a single-fibre model that describes the axial response of individual fibres, with network models that assemble individual fibre properties into overall network behaviour. In particular, we consider four different network models, including the affine, three-chain, eight-chain, and micro-sphere models, which employ different assumptions about network structure and kinematics. We systematically investigate the ability of these models to describe the mechanical response of athermal collagen and fibrin networks by comparing model predictions with experimental data. We test how each model captures network behaviour under three different loading conditions: uniaxial tension, simple shear, and combined tension and shear. We find that the affine and three-chain models can accurately describe both the axial and shear behaviour, whereas the eight-chain and micro-sphere models fail to capture the shear response, leading to unphysical zero shear moduli at infinitesimal strains. Our study is the first to systematically investigate the applicability of popular network models for describing the macroscopic behaviour of athermal fibrous networks, offering insights for selecting efficient models that can be used for large-scale, finite-element simulations of athermal networks.

Structure-Elastic Properties Relationships in Gelling Carrageenans

Gelling carrageenans are polysaccharides extracted from the Gigartinales order of red algae. These are additives used essentially in the food industry for texturizing, stabilizing or gelling various formulations. Although a consensual gel mechanism has been reached which encompasses a coil-to-helix transition followed by the self-assembling of helices in a network, the structure–elastic relationships in the network are still to be clearly established. This paper reviews the reports in which carrageenan gel structures have been systematically compared with gel elastic properties. The focus is on the sizes documented for structural units, such as strands, aggregates, voids or network meshes, as well as on the reported linear and nonlinear elastic characteristics. The insufficient rationalization of carrageenan gel elasticity by models which take on board mechanically relevant structural features is underlined. After introducing selected linear and nonlinear elastic models, preliminary results comparing such models to structural and rheological data are presented. In particular, the concentration scaling of the strain hardening exhibited by two types of carrageenan gels is discussed.

Strain localization and yielding dynamics in disordered collagen networks

Collagen is the most abundant extracellular-matrix protein found in mammals and the main structural and load-bearing element of connective tissues. Collagen networks show remarkable strain-stiffening, which tunes the mechanical functions of tissues and regulates cell behaviours. Linear and non-linear mechanics of in vitro disordered collagen networks have been widely studied using rheology for a range of self-assembly conditions in recent years. However, the correlation between the onset of macroscopic network failure and local deformations is not well understood in these systems. Here, using shear rheology and in situ high-resolution boundary imaging, we study the yielding dynamics of in vitro reconstituted networks of uncrosslinked type-I collagen. We find that in the non-linear regime, the differential shear modulus (K) of the network initially increases with applied strain and then begins to drop as the network starts to yield beyond a critical strain (yield strain). Measurement of the local velocity profile using colloidal tracer particles reveals that beyond the peak of K, strong strain-localization and slippage between the network and the rheometer plate sets in that eventually leads to a detachment. We generalize this observation for a range of collagen concentrations, applied strain ramp rates, as well as, different network architectures obtained by varying the polymerization temperature. Furthermore, using a continuum affine network model, we map out a state diagram showing the dependence of yield-stain and -stress on the microscopic network parameters. Our findings can have broad implications in tissue engineering and designing highly resilient biological scaffolds.

Continuum elastic models for force transmission in biopolymer gels

We review continuum elastic models for the transmission of both external forces and internal active cellular forces in biopolymer gels, and relate them to recent experiments. Rather than being exhaustive, we focus on continuum elastic models for small affine deformations and intend to provide a systematic continuum method and some analytical perspectives on the study of force transmission in biopolymer gels. We start from a very brief review of the nonlinear mechanics of individual biopolymers and a summary of constitutive models for the nonlinear elasticity of biopolymer gels. We next show that the simple 3-chain model can give predictions that fit well the shear experiments of some biopolymer gels, including the effects of strain-stiffening and negative normal stress. We then review continuum models for the transmission of internal active forces that are induced by a spherically contracting cell embedded in a three-dimensional biopolymer gel. Various scaling regimes for the decay of cell-induced displacements are identified for linear isotropic and anisotropic materials, and for biopolymer gels with nonlinear compressive-softening and strain-stiffening elasticity, respectively. After that, we present (using an energetic approach) the generic and unified continuum theory proposed in [D. Ben-Yaakov et al., Soft Matter, 2015, 11, 1412] about how the transmission of forces in the biogel matrix can mediate long-range interactions between cells with mechanical homeostasis. We show the predictions of the theory in a special hexagonal multicellular array, and relate them to recent experiments. Finally, we conclude this paper with comments on the limitations and outlook of continuum modeling, and highlight the need for complementary theoretical approaches, such as discrete network simulations, to force transmission in biopolymer gels and phenomenological active gel theories for multicellular systems.

Strain Stiffening and Negative Normal Force of Agarose Hydrogels

Inspired by the specific strain stiffening and negative normal force phenomena in several biological networks, herein, we show strain stiffening and negative normal force in agarose hydrogels. We use both pre-strain and strain amplitude sweep protocols in dynamic rheological measurements where the gel slip was suppressed by the in situ gelation in the cross-hatched parallel plate rheometer geometry. Within the stiffening region, we show the scaling relation for the differential modulus K ∝ σ1, where σ is stress. The strain at the onset of stiffening is almost constant throughout the concentration range. The gels show negative apparent normal stress difference when sheared as a result of the gel contraction. The pore size of the hydrogel is large enough to allow water to move with respect to the network to balance the pressure difference caused by the hoop stress. The rheological analysis together with scanning electron microscopy suggests that the agarose gels can be described using subisostatic athermal network models where the connectivity dictates the stiffening behavior. Therefore, the simple agarose gels appear to capture several of the viscoelastic properties, which were previously thought to be characteristic to biological protein macromolecules.

Exploiting Nonlinear Elasticity for Anomalous Magnetoresponsive Stiffening

A paradigm for enhanced magnetorheological elastic materials is introduced and experimentally established. We show that a nonlinearly stiffening polymer matrix can be exploited to achieve anomalous magneto-elastomer stiffening exceeding standard magneto-elastomer theory and experiment in terms of percentage stiffness change and sensitivity to applied magnetic flux. Using a model system of a semiflexible fibrin network embedded with micron sized carbonyl iron particles, we demonstrate that even at a modest particle volume fraction (0.5-4%), a coupling between the magnetically interacting dipoles and a strain-stiffening polymer mesh provides previously unexplored opportunities for material design. Our experiments indicate that confined particles within the fibrin network internally tension and stiffen the polymer mesh when an external field is applied, resulting in a field-dependent stiffening response from the polymer mesh that superposes with the magnetic interparticle interactions.

Rapid local compression in active gels is caused by nonlinear network response

The actin cytoskeleton in living cells generates forces in conjunction with myosin motor proteins to directly and indirectly drive essential cellular processes. The semiflexible filaments of the cytoskeleton can respond nonlinearly to the collective action of motors. We here investigate mechanics and force generation in a model actin cytoskeleton, reconstituted in vitro, by observing the response and fluctuations of embedded micron-scale probe particles. Myosin mini-filaments can be modeled as force dipoles and give rise to deformations in the surrounding network of cross-linked actin. Anomalously correlated probe fluctuations indicate the presence of rapid local compression or draining of the network that emerges in addition to the ordinary linear shear elastic (incompressible) response to force dipoles. The anomalous propagation of compression can be attributed to the nonlinear response of actin filaments to the microscopic forces, and is quantitatively consistent with motor-generated large-scale stiffening of the gels.

The elastic properties and deformation mechanisms of actin filament networks crosslinked by filamins

As a substructure of cell cytoskeleton, the crosslinked actin filament networks (CAFNs) play a major role in different cell functions, however, the elastic properties and the deformation mechanisms of CAFNs still remain to be understood. In this paper, a novel three-dimensional (3D) finite element (FE) model has been developed to mimic the mechanical properties of actin filament (F-actin) networks crosslinked by filamin A (FLNA). The simulation results indicate that although the Young's modulus of CAFNs varies in different directions for each random model, the statistical mean value is in-plane isotropic. The crosslinking density and the actin filament volume fraction are found to strongly affect the in-plane shear modulus of CAFNs. The simulation results agree well with the relevant experimental results. In addition, an L-shaped cantilever beam model has been developed for dimensional analysis on the shear stiffness of CAFNs and for quantifying the deformation mechanisms. It has been demonstrated that the in-plane shear modulus of CAFNs is mainly dominated by FLNA (i.e., cross-linkers), and that the bending and torsion deformations of FLNA have almost the same contribution to the stiffness of CAFNs. It has also been found that the stiffness of CAFNs is almost insensitive to the variation of the Poisson's ratios of FLNA and actin filament in the range from 0.29 to 0.499.

Experimental study of the Poynting effect in a soft unidirectional fiber-reinforced material under simple shear

The aim of this work was to investigate the shear and lateral normal responses of a soft unidirectional fiber-reinforced material subjected to simple shear. The Poynting effect was also investigated. Soft composites were manufactured from a flexible adhesive reinforced by a single family of parallel and continuous fibers of nylon. Specimens with fibers oriented at an angle (-45°, 0°, 45° and 90°) with respect to the applied shear force were tested. A simple shear test apparatus was developed to measure shear and normal forces simultaneously. A standard reinforcing model based on strain-energy density function was used to verify the mechanical behavior of the soft composite with different fiber orientation. Results showed that the initial stiffness of the composite with fibers oriented at -45° and 45° was approximately the same and was higher than those at 0° and 90°. Also, there was no significant difference between values of initial stiffness for angles of 0° and 90° and the neat matrix. The effect of the stretching resistance of the fibers was more pronounced for fibers oriented at 45° and 90°. There was no Poynting effect for the neat matrix or for the composite with fibers at 0° while positive and negative Poynting effects were observed for fibers oriented at -45° and 45° (and 90°), respectively. The standard reinforcing model was only verified for a limited range of amount of shear due to composite failure. Fiber debonding and fiber buckling were observed in the composites with fibers oriented at 45° (and 90°) and -45°, respectively, at large deformations.

Biomimetic Strain-Stiffening Self-Assembled Hydrogels

Supramolecular structures with strain-stiffening properties are ubiquitous in nature but remain rare in the lab. Herein, we report on strain-stiffening supramolecular hydrogels that are entirely produced through the self-assembly of synthetic molecular gelators. The involved gelators self-assemble into semi-flexible fibers, which thereby crosslink into hydrogels. Interestingly, these hydrogels are capable of stiffening in response to applied stress, resembling biological intermediate filaments system. Furthermore, strain-stiffening hydrogel networks embedded with liposomes are constructed through orthogonal self-assembly of gelators and phospholipids, mimicking biological tissues in both architecture and mechanical properties. This work furthers the development of biomimetic soft materials with mechanical responsiveness and presents potentially enticing applications in diverse fields, such as tissue engineering, artificial life, and strain sensors.

Dehydropeptide-based plasmonic magnetogels: a supramolecular composite nanosystem for multimodal cancer therapy

Supramolecular hydrogels are highly promising candidates as biomedical materials owing to their wide array of properties, which can be tailored and modulated. Additionally, their combination with plasmonic/magnetic nanoparticles to form plasmonic magnetogels further improves their potential in biomedical applications through the combination of complementary strategies, such as photothermia, magnetic hyperthermia, photodynamic therapy and magnetic-guided drug delivery. Here, a new dehydropeptide hydrogelator, Npx-l-Met-Z-ΔPhe-OH, was developed and combined with two different plasmonic/magnetic nanoparticle architectures, i.e., core/shell manganese ferrite/gold nanoparticles and gold-decorated manganese ferrite nanoparticles with ca. 55 nm and 45 nm sizes, respectively. The magnetogels were characterized via HR-TEM, FTIR spectroscopy, circular dichroism and rheological assays. The gels were tested as nanocarriers for a model antitumor drug, the natural compound curcumin. The incorporation of the drug in the magnetogel matrices was confirmed through fluorescence-based techniques (FRET, fluorescence anisotropy and quenching). The curcumin release profiles were studied with and without the excitation of the gold plasmon band. The transport of curcumin from the magnetogels towards biomembrane models (small unilamellar vesicles) was assessed via FRET between the fluorescent drug and the lipid probe Nile Red. The developed magnetogels showed promising results for photothermia and photo-triggered drug release. The magnetogels bearing gold-decorated nanoparticles showed the best photothermia properties, while the ones containing core/shell nanoparticles had the best photoinduced curcumin release.

Rheological properties and failure of alginate hydrogels with ionic and covalent crosslinks

Polysaccharide-based hydrogels are being used in a wide variety of applications ranging from tissue engineering to food products due to their biocompatibility and the ease of gel formation. In real-life applications, hydrogels can undergo large strain deformation, which may result in structural damage leading to failure. Here, we report the nonlinear rheological properties and failure behavior of alginate hydrogels, a class of polysaccharide hydrogels, synthesized via ionic and covalent crosslinking. Gels with ionic crosslinks or ionic alginate hydrogels are prepared by addition of Ca2+ ions in the aqueous solution of sodium alginate, and the covalently crosslinked alginate gels or chemical alginate hydrogels are obtained via amidation reactions. Because of their structural differences, ionic and chemical alginate hydrogels display different scattering profiles captured by using small angle X-ray scattering (SAXS) technique. Both ionic and chemical alginate hydrogels exhibit strain stiffening behavior when subjected to large amplitude oscillatory shear (LAOS) and the strain-stiffening behavior is accompanied by negative normal stress. A custom-built cavitation rheometer has been utilized to probe the local failure behavior of these gels. The cavitation rheometry captures different defect growth or fracture mechanism in ionic versus chemical alginate hydrogels, even if these two types of gels have a similar linear elastic modulus. Based on the critical pressure for gel fracture, we have provided an estimate of the critical energy release rate.

A mesoscopic theory to describe the flexibility regulation in F-actin networks: An approach of phase transitions with nonlinear elasticity

The synthetic actin network arouses great interest as bio-material due to its soft and wet nature that mimics many biological scaffolding structures. Inside the cell, the actin network is regulated by tens of actin-binding proteins (ABP's), which make for a highly complex system with several emergent behaviors. In particular, calponin is an ABP that was identified as an actin stabiliser, but whose mechanism is still poorly understood. Recent experiments using an in vitro model system of cross-linked actin with calponin and large deformation bulk rheology, found that networks with exhibited a delayed onset and were able to withstand a higher maximal strain before softening. In this work, we show that at network scale the actin network with calponin furthermore the reduction of the persistence length allows: (i) The reduction in the network pre-strain. (ii) The increment of the crosslinks adhesion energy. We verify these effects theoretically using nonlinear continuum mechanics for the semiflexible and crosslinked network. In addition, the alterations over the microstructure are described by the definition of an interaction parameter Γ according the formalism of Landau for phase transitions. According to this model we demonstrates that the interaction parameter can describe the experimental observations following a scaling exponent as Γ~|c-c_cr|^1/2, where c is the ratio between concentration of calponin and actin. This result provides interesting feedback to improve our understanding of several mechano-biological pathways.

Normal stress anisotropy and marginal stability in athermal elastic networks

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 σzz on long (compressible) timescales, crossing over to the first normal stress difference, N1 = σxx - σzz at short (incompressible) times. While the behavior of N1 is well understood for isotropic viscoelastic materials undergoing affine shear deformation, biopolymer networks are often anisotropic and deform nonaffinely. Here, we numerically study the normal stresses that arise under shear in subisostatic, athermal semiflexible polymer networks. We show that such systems exhibit strong deviations from affine behavior and that these anomalies are controlled by a rigidity transition as a function of strain.

Inelastic behaviour of cellulose microfibril networks

Cellulose microfibrils (CMF) are a unique class of shape anisotropic bio-nanomaterials, already finding many applications in diverse fields owing to their advantageous material properties and abundant availability. The rich non-linear mechanical behaviour of CMF networks has been under-studied due to the complex nature of this system, being influenced by many factors such as strong inter-fibril interactions, a heterogeneous microstructure, and process conditions. In this work, we systematically explore the non-linear rheological behaviour of these networks using a CMF model system with controlled process conditions and fibril interactions. The microfibrils were dispersed in dimethyl sulfoxide to minimise the attractive van der Waals interactions and thereby also the network heterogeneity. We show that the networks exhibit a transition with increasing shear stress from a predominantly elastic to a plastic deformation where they undergo softening. We find that the network stiffness and plasticity are dependent on the loading rate. Finally, we observed that the networks regain their original viscoelastic moduli on cessation of shear. These findings form a basis towards understanding and ultimately modelling the mechanics of CMF networks, which is a prerequisite for the rational design of novel bio-based materials.

Normal Stresses, Contraction, and Stiffening in Sheared Elastic Networks

When elastic solids are sheared, a nonlinear effect named after Poynting gives rise to normal stresses or changes in volume. We provide a novel relation between the Poynting effect and the microscopic Grüneisen parameter, which quantifies how stretching shifts vibrational modes. By applying this relation to random spring networks, a minimal model for, e.g., biopolymer gels and solid foams, we find that networks contract or develop tension because they vibrate faster when stretched. The amplitude of the Poynting effect is sensitive to the network's linear elastic moduli, which can be tuned via its preparation protocol and connectivity. Finally, we show that the Poynting effect can be used to predict the finite strain scale where the material stiffens under shear.

Poynting and reverse Poynting effects in soft materials

In 1909, J. H. Poynting conducted a series of experiments on metal wires and found that loaded wires lengthen when twisted. Thus to maintain a constant length in such experiments, a compressive axial force would need to be applied at the ends of the specimen. This is the classical (positive) Poynting effect. Another example of such an effect arises when a soft material specimen is being axially sheared or rotated between two platens. The necessity to apply a normal force in order to maintain the relative distance between the platens is also often referred to as a Poynting-type effect. Both effects are inherently nonlinear phenomena. In recent papers, experimental data on the Poynting effect in soft solids have been reported. The seminal paper by Janmey et al. describes shearing experiments on hydrogels, impregnated with scleroproteins such as collagen and fibrin. It was shown that positive and negative (reverse) Poynting effects could occur. In this and subsequent papers by several authors, the microstructure of reinforced biogels involving semi-flexible filaments embedded in a soft matrix was exploited to examine the character of the normal stresses. The purpose of the present paper is to describe and review an alternative approach using the macroscopic phenomenological theory of hyperelasticity based on nonlinear continuum mechanics. Our aim is to demonstrate that such a theory can be used in a very transparent way to predict the occurrence of both positive and negative Poynting effects in anisotropic soft fibrous materials. It will be seen that material anisotropy plays a key role in the analysis.

Large Network Swelling and Solvent Redistribution Are Necessary for Polymer Gels to Show Negative Normal Stress

We use a statistical thermodynamic model to theoretically predict the normal stress of a polymer gel, which is twisted by a rotational rheometer in the parallel plate geometry. This theory predicts that the normal stress of the gel is positive at a short time scale. At longer time, the normal stress decreases significantly due to the redistribution of solvent, but is still positive when the polymer network is in the neo-Hookean regime. The normal stress becomes negative when the polymer network is greatly prestretched in the swelling process because large tension is generated by shearing already prestretched polymers. Our theory predicts the phase diagram with respect to the direction of normal stress as functions of the aspect ratio and swelling ratio of gels. Experimental tests on this prediction may advance our understanding of the physical mechanisms involved in the negative normal stress generated by polymer gels.

Effects of disorder and chain stiffening on the elasticity of flexible polymer networks

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 two-dimensional 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 nonaffine contribution to the shear modulus is negligible, i.e., the Born approximation is satisfactory, and (ii) the shear modulus is dominated by the contribution originating from equilibrium chain tensions. Moreover, mechanical equilibration at the nodes induces specific correlations between the end-to-end distances and contour lengths of chain segments, which must be properly accounted for to construct reasonable estimates of chain pressure and shear modulus.

Theory of Semiflexible Filaments and Networks

We briefly review the recent developments in the theory of individual semiflexible filaments, and of a crosslinked network of such filaments, both permanent and transient. Starting from the free energy of an individual semiflexible chain, models on its force-extension relation and other mechanical properties such as Euler buckling are discussed. For a permanently crosslinked network of filaments, theories on how the network responds to deformation are provided, with a focus on continuum approaches. Characteristic features of filament networks, such as nonlinear stress-strain relation, negative normal stress, tensegrity, and marginal stability are discussed. In the new area of transient filament network, where the crosslinks can be dynamically broken and re-formed, we show some recent attempts for understanding the dynamics of the crosslinks, and the related rheological properties, such as stress relaxation, yield stress and plasticity.

Ion pairing controls rheological properties of "processionary" polyelectrolyte hydrogels

We demonstrated recently that polyelectrolytes with cationic moieties along the chain and a single anionic head are able to form physical hydrogels due to the reversible nature of the head-to-body ionic bond. Here we generate a variety of such polyelectrolytes with various cationic moieties and counterion combinations starting from a common polymeric platform. We show that the rheological properties (shear modulus, critical strain) of the final hydrogels can be modulated over three orders of magnitude depending on the cation/anion pair. Our data fit remarkably well within a scaling model involving a supramolecular head-to-tail single file between cross-links, akin to the behaviour of pine-processionary caterpillar. This model allows the quantitative measure of the amount of counterion condensation from standard rheology procedure.