Linear viscoelastic properties of transient networks formed by associating polymers with multiple stickers

Non-Maxwellian viscoelastic stress relaxations in soft matter

Viscoelastic stress relaxation is a basic characteristic of soft matter systems such as colloids, gels, and biological networks. Although the Maxwell model of linear viscoelasticity provides a classical description of stress relaxation, it is often not sufficient for capturing the complex relaxation dynamics of soft matter. In this Tutorial, we introduce and discuss the physics of non-Maxwellian linear stress relaxation as observed in soft materials, the ascribed origins of this effect in different systems, and appropriate models that can be used to capture this relaxation behavior. We provide a basic toolkit that can assist the understanding and modeling of the mechanical relaxation of soft materials for diverse applications.

Optimizing the accuracy of viscoelastic characterization with AFM force-distance experiments in the time and frequency domains

Atomic Force Microscopy (AFM) force-distance (FD) experiments have emerged as an attractive alternative to traditional micro-rheology measurement techniques owing to their versatility of use in materials of a wide range of mechanical properties. Here, we show that the range of time dependent behaviour which can reliably be resolved from the typical method of FD inversion (fitting constitutive FD relations to FD data) is inherently restricted by the experimental parameters: sampling frequency, experiment length, and strain rate. Specifically, we demonstrate that violating these restrictions can result in errors in the values of the parameters of the complex modulus. In the case of complex materials, such as cells, whose behaviour is not specifically understood a priori, the physical sensibility of these parameters cannot be assessed and may lead to falsely attributing a physical phenomenon to an artifact of the violation of these restrictions. We use arguments from information theory to understand the nature of these inconsistencies as well as devise limits on the range of mechanical parameters which can be reliably obtained from FD experiments. The results further demonstrate that the nature of these restrictions depends on the domain (time or frequency) used in the inversion process, with the time domain being far more restrictive than the frequency domain. Finally, we demonstrate how to use these restrictions to better design FD experiments to target specific timescales of a material's behaviour through our analysis of a polydimethylsiloxane (PDMS) polymer sample.

Linear Dynamic Viscoelasticity of Dual Cross-Link Poly(Vinyl Alcohol) Hydrogel with Determined Borate Ion Concentration

We investigated the linear dynamic viscoelasticity of dual cross-link (DC) poly(vinyl alcohol) (PVA) (DC-PVA) hydrogels with permanent and transient cross-links. The concentrations of incorporated borate ions to form transient cross-links in the DC-PVA hydrogels (CBIN) were determined by the azomethine-H method. The dynamic viscoelasticity of the DC-PVA hydrogel cannot be described by a simple sum of the dynamic viscoelasticity of the PVA gel with the same permanent cross-link concentration and the PVA aqueous solution with the same borate ion concentration (CB = CBIN) as in the DC-PVA gel. The DC-PVA hydrogel exhibited a considerably higher relaxation strength, indicating that the introduction of permanent cross-links into temporary networks increases the number of viscoelastic chains with finite relaxation times. In contrast, the relaxation frequency (ωc) (given by the frequency at the maximum of loss modulus) for the DC-PVA hydrogel was slightly lower but comparable to that for a dilute PVA solution with the same CB. This signifies that the relaxation dynamics of the DC-PVA hydrogels is essentially governed by the lifetime of an interchain transient cross-link (di-diol complex of boron). The effect of permanent cross-linking on the relaxation dynamics was observed in the finite broadening of the relaxation-time distribution in the long time region.

Multiaxial Stress Relaxation of Dual-Cross-Link Poly(vinyl alcohol) Hydrogels

Dual-cross-link hydrogels, which are composed of poly(vinyl alcohol) chains cross-linked by covalent bonds and transient physical bonds via borate ions, have received considerable interest due to the high extensibility as well as the self-healing properties. The dual-cross-link gels exhibit pronounced stress relaxation after the imposition of step strain, reflecting the breaking dynamics of temporary chains which sustain the initial stress. Multiaxial stress relaxation experiments using equibiaxial, planar, and uniaxial stretching with various degrees of strain unambiguously validate the time-strain separability in relaxation stress for a general type of deformation. The time effect is satisfactorily described by an existing model, while the strain effect is well approximated by the neo-Hookean model. Furthermore, the magnitude of total stress relaxation to the initial stress is constant, independently of the type and degree of imposed strain. These simple features provide an important basis for formulating the constitutive models of the dual-cross-link gels and for molecular design of hydrogels with transient physical cross-links.

Physical gels of poly(vinylamine) by thermal curing

Physical gels are a versatile class of materials which can find application in sensors, electrochemistry, biomedicine or rheological modifiers. Herein, we present a hydrogen-bonded physical gel which is based on the interaction between phenylcarbonate telechelic poly(ethylene glycol) (PEG-PC) and poly(vinyl amine-co-acetamide) (p(VAm-co-VAA)). The critical gelation concentration was found to be 10 wt% by rheology and NMR. UV-vis spectroscopy and dynamic light scattering reveal the formation of aggregates in the gel. Rheology and differential scanning calorimetry (DSC) was used to show the effect of thermal curing on the mechanical properties of the physical gel.

Repeated heating cooling cycles can anneal inhomogeneity in physical gels and significantly improve their mechanical properties.

Rouse model with transient intramolecular contacts on a timescale of seconds recapitulates folding and fluctuation of yeast chromosomes

DNA folding and dynamics along with major nuclear functions are determined by chromosome structural properties, which remain, thus far, elusive in vivo. Here, we combine polymer modeling and single particle tracking experiments to determine the physico-chemical parameters of chromatin in vitro and in living yeast. We find that the motion of reconstituted chromatin fibers can be recapitulated by the Rouse model using mechanical parameters of nucleosome arrays deduced from structural simulations. Conversely, we report that the Rouse model shows some inconsistencies to analyze the motion and structural properties inferred from yeast chromosomes determined with chromosome conformation capture techniques (specifically, Hi-C). We hence introduce the Rouse model with Transient Internal Contacts (RouseTIC), in which random association and dissociation occurs along the chromosome contour. The parametrization of this model by fitting motion and Hi-C data allows us to measure the kinetic parameters of the contact formation reaction. Chromosome contacts appear to be transient; associated to a lifetime of seconds and characterized by an attractive energy of -0.3 to -0.5 kBT. We suggest attributing this energy to the occurrence of histone tail-DNA contacts and notice that its amplitude sets chromosomes in 'theta' conditions, in which they are poised for compartmentalization and phase separation.

Harnessing entropy to enhance toughness in reversibly crosslinked polymer networks

Reversible crosslinking is a design paradigm for polymeric materials, wherein they are microscopically reinforced with chemical species that form transient crosslinks between the polymer chains. Besides the potential for self-healing, recent experimental work suggests that freely diffusing reversible crosslinks in polymer networks, such as gels, can enhance the toughness of the material without substantial change in elasticity. This presents the opportunity for making highly elastic materials that can be strained to a large extent before rupturing. Here, we employ Gaussian chain theory, molecular simulation, and polymer self-consistent field theory for networks to construct an equilibrium picture for how reversible crosslinks can toughen a polymer network without affecting its elasticity. Maximisation of polymer entropy drives the reversible crosslinks to bind preferentially near the permanent crosslinks in the network, leading to local molecular reinforcement without significant alteration of the network topology. In equilibrium conditions, permanent crosslinks share effectively the load with neighbouring reversible crosslinks, forming multi-functional crosslink points. The network is thereby globally toughened, while the linear elasticity is left largely unaltered. Practical guidelines are proposed to optimise this design in experiment, along with a discussion of key kinetic and timescale considerations.

The Effects of the Interplay between Motor and Brownian Forces on the Rheology of Active Gels

Active gels perform key mechanical roles inside the cell, such as cell division, motion, and force sensing. The unique mechanical properties required to perform such functions arise from the interactions between molecular motors and semiflexible polymeric filaments. Molecular motors can convert the energy released in the hydrolysis of ATP into forces of up to piconewton magnitudes. Moreover, the polymeric filaments that form active gels are flexible enough to respond to Brownian forces but also stiff enough to support the large tensions induced by the motor-generated forces. Brownian forces are expected to have a significant effect especially at motor activities at which stable noncontractile in vitro active gels are prepared for rheological measurements. Here, a microscopic mean-field theory of active gels originally formulated in the limit of motor-dominated dynamics is extended to include Brownian forces. In the model presented here, Brownian forces are included accurately, at real room temperature, even in systems with high motor activity. It is shown that a subtle interplay, or competition, between motor-generated forces and Brownian forces has an important impact on the mass transport and rheological properties of active gels. The model predictions show that at low frequencies the dynamic modulus of active gels is determined mostly by motor protein dynamics. However, Brownian forces significantly increase the breadth of the relaxation spectrum and can affect the shape of the dynamic modulus over a wide frequency range even for ratios of motor to Brownian forces of more than a hundred. Since the ratio between motor and Brownian forces is sensitive to ATP concentration, the results presented here shed some light on how the transient mechanical response of active gels changes with varying ATP concentration.

The effect of shear history on urea containing gliadin solutions

Currently, a substantial amount of research is devoted to gluten bioplastics. A promising processing route towards composites and films uses solutions of reduced gliadin. The addition of sufficient urea allows the preparation of highly concentrated gliadin solutions without an anomalous rheology. This is investigated in this paper by thixotropy experiments on gliadin solutions. These solutions show a balance between structural build-up due to molecular interactions and structural break-down induced by shear flow. Because of this, such protein solutions should be prepared with great caution. To assure a rheology suitable for processing, a shear history and a sufficient amount of added urea to disrupt molecular interactions are crucial.

Mechanical response of transient telechelic networks with many-part stickers

A central question in soft matter is understanding how several individual, weak bonds act together to produce collective interactions. Here, gel-forming telechelic polymers with multiple stickers at each chain end are studied through Brownian dynamics simulations to understand how collective interaction of the bonds affects mechanical response of the gels. These polymers are modeled as finitely extensible dumbbells using an explicit tau-leap algorithm and the binding energy of these associations was kept constant regardless of the number of stickers. The addition of multiple bonds to the associating ends of telechelic polymers increases or decreases the network relaxation time depending on the relative kinetics of association but increases both shear stress and extensional viscosity. The relationship between the rate of association and the Rouse time of dangling chains results in two different regimes for the equilibrium stress relaxation of associating physical networks. In case I, a dissociated dangling chain is able to fully relax before re-associating to the network, resulting in two characteristic relaxation times and a non-monotonic terminal relaxation time with increasing number of bonds per polymer endgroup. In case II, the dissociated dangling chain is only able to relax a fraction of the way before it re-attaches to the network, and increasing the number of bonds per endgroup monotonically increases the terminal relaxation time. In flow, increasing the number of stickers increases the steady-state shear and extensional viscosities even though the overall bond kinetics and equilibrium constant remain unchanged. Increased dissipation in the simulations is primarily due to higher average chain extension with increasing bond number. These results indicate that toughness and dissipation in physically associating networks can both be increased by breaking single, strong bonds into smaller components.

Supramolecular polymer hydrogels induced by host-guest interactions with di-[cyclobis(paraquat-p-phenylene)] cross-linkers: from molecular complexation to viscoelastic properties

Supramolecular polymer networks have been designed on the basis of a π-electron donor/acceptor complex: naphthalene (N)/cyclobis(paraquat-p-phenylene) (CBPQT⁴⁺ = B). For this purpose, a copolymer of N,N-dimethylacrylamide P(DMA-N1), lightly decorated with 1 mol% of naphthalene pendant groups, has been studied in semi-dilute un-entangled solution in the presence of di-CBPQT⁴⁺ (BB) crosslinker type molecules. While calorimetric experiments demonstrate the quantitative binding between N and B groups up to 60 °C, the introduction of BB crosslinkers into the polymer solution gives rise to gel formation above the overlap concentration. From a comprehensive investigation of viscoelastic properties, performed at different concentrations, host/guest stoichiometric ratios and temperatures, the supramolecular hydrogels are shown to follow a Maxwellian behavior with a strong correlation of the plateau modulus and the relaxation time with the effective amount of interchain cross-linkers and their dissociation dynamics, respectively. The calculation of the dissociation rate constant of the supramolecular complex, by extrapolation of the relaxation time of the network back to the beginning of the gel regime, is discussed in the framework of theoretical and experimental works on associating polymers.

One-Step Fabrication of pH-Responsive Membranes and Microcapsules through Interfacial H-Bond Polymer Complexation

Biocompatible microencapsulation is of widespread interest for the targeted delivery of active species in fields such as pharmaceuticals, cosmetics and agro-chemistry. Capsules obtained by the self-assembly of polymers at interfaces enable the combination of responsiveness to stimuli, biocompatibility and scaled up production. Here, we present a one-step method to produce in situ membranes at oil-water interfaces, based on the hydrogen bond complexation of polymers between H-bond acceptor and donor in the oil and aqueous phases, respectively. This robust process is realized through different methods, to obtain capsules of various sizes, from the micrometer scale using microfluidics or rotor-stator emulsification up to the centimeter scale using drop dripping. The polymer layer exhibits unique self-healing and pH-responsive properties. The membrane is viscoelastic at pH = 3, softens as pH is progressively raised, and eventually dissolves above pH = 6 to release the oil phase. This one-step method of preparation paves the way to the production of large quantities of functional capsules.

Celebrating Soft Matter's 10th Anniversary: chain configuration and rate-dependent mechanical properties in transient networks

Numerical solution of a coupled set of Smoluchowski convection-diffusion equations of associating polymers modelled as finitely extensible dumbbells enables computation of time-dependent end-to-end distributions for bridged, dangling, and looped chains in three dimensions as a function of associating end-group kinetics. Non-monotonic flow curves which can lead to flow instabilities during shear flow result at low equilibrium constant and high association rate from two complementary phenomena: a decrease in the fraction of elastically active chains with increasing shear rate and non-monotonic extension in the population of elastically active chains. Chain tumbling leads to reformation of bridges, resulting in an increased fraction of bridged chains at high Deborah number and significant reduction in the average bridge chain extension. In the start-up of steady shear, force-activated chain dissociation and chain tumbling cause both stress overshoot and stress ringing behaviour prior to reaching steady state stress values. During stress relaxation following steady shear, chain kinetics and extension mediate both the number of relaxations and the length of time required for system relaxation. While at low association rate relaxation is limited by the relaxation of dangling chains and the rate of dangling chain formation, at high association rate coupling of dangling and bridged chains leads to simultaneous relaxation of all chains due to a dynamic equilibrium between dangling and bridged states.

Anomalous self-diffusion and sticky Rouse dynamics in associative protein hydrogels

Natural and synthetic materials based on associating polymers possess diverse mechanical behavior, transport properties and responsiveness to external stimuli. Although much is known about their dynamics on the molecular and macroscopic level, knowledge of self-diffusive dynamics of the network-forming constituents remains limited. Using forced Rayleigh scattering, anomalous self-diffusion is observed in model associating protein hydrogels originating from the interconversion between species that diffuse in both the molecular and associated state. The diffusion can be quantitatively modeled using a two-state model for polymers in the gel, where diffusivity in the associated state is critical to the super diffusive behavior. The dissociation time from bulk rheology measurements was 2-3 orders of magnitude smaller than the one measured by diffusion, because the former characterizes submolecular dissociation dynamics, whereas the latter depicts single protein molecules completely disengaging from the network. Rheological data also show a sticky Rouse-like relaxation at long times due to collective relaxation of large groups of proteins, suggesting mobility of associated molecules. This study experimentally demonstrates a hierarchy of relaxation processes in associating polymer networks, and it is anticipated that the results can be generalized to other associative systems to better understand the relationship of dynamics among sticky bonds, single molecules, and the entire network.

The role of filament length, finite-extensibility and motor force dispersity in stress relaxation and buckling mechanisms in non-sarcomeric active gels

After relaxing some assumptions we apply a single-chain mean-field mathematical model recently introduced [RSC Adv. (2014)] to describe the role of molecular motors in the mechanical properties of active gels. The model allows physics that are not available in models postulated on coarser levels of description. Moreover it proposes a level of description that allows the prediction of observables at time scales too difficult to achieve in multi-chain simulations for realistic filament lengths and densities. We model the semiflexible filaments that compose the active gel as bead-spring chains; molecular motors are accounted for by using a mean-field approach, in which filaments undergo transitions of one motor attachment state depending on the state of the probe filament. The level of description includes the end-to-end distance and attachment state of the filaments, and the motor-generated forces, as stochastic state variables which evolve according to a proposed differential Chapman-Kolmogorov equation. The motor-generated forces are drawn from a stationary distribution of motor stall forces. We consider bead-spring chains with multiple beads, explore the effect of finite-extensibility of the strands and incorporate into the model motor force distributions that have been measured experimentally. The model can no longer be solved analytically but is amenable to numerical simulation. This version of the model allows a more quantitative description of buckling dynamics [Lenz et. al. PRL, 2012, 108, 238107] and the dynamic modulus of active gels. The effect of finite extensibility of the filament strands on the dynamic modulus was also found to be in agreement with the microrheology experiments of Mizuno et. al., [Science, 2007, 315, 370-373].

Stress-Strain Relationship of Highly Stretchable Dual Cross-Link Gels: Separability of Strain and Time Effect

We studied the stress-strain relation of model dual cross-link gels having permanent cross-links and transient cross-links over an unusually wide range of extension ratios λ and strain rates ϵ̇ (or time t = (λ - 1)/ϵ̇). We propose a new analysis method and separate the stress into strain- and time-dependent terms. The strain-dependent term is derived from rubber elasticity, while the time-dependent term is due to the failure of transient cross-links and can be represented as a time-dependent shear modulus which shows the same relaxation as in small strain. The separability is applicable except for the strain stiffening regimes resulting from the finite extensibility of polymer chains. This new analysis method should have a wide applicability not only for hydrogels but also for other highly viscoelastic soft solids such as soft adhesives or living tissues.