Inverse mechanical-swelling coupling of a highly deformed double-network gel.

Mechanical behaviors of a polymer gel are coupled with its swelling behavior. It has been known that typical hydrogels display extension-induced swelling and drying-induced stiffening, called normal mechanical-swelling coupling. In this study, we experimentally found that highly extended double-network (DN) hydrogels exhibit abnormal inverse mechanical-swelling coupling such as extension-induced deswelling and drying-induced softening. We established theoretical hyperelastic and swelling models that reproduced all the complicated mechanical and swelling trends of the highly deformed DN hydrogels. From these theoretical analyses, it is considered that the inverse mechanical-swelling coupling of a DN gel is derived from the extreme nonlinear elasticity of its first network at its ultimate deformation state. These findings contribute toward the understanding of the mechanics of rubber-like materials up to their ultimate deformation and fracture limit.

A simple microswimmer model inspired by the general equation for nonequilibrium reversible-irreversible coupling.

A simple mean-field microswimmer model is presented. The model is inspired by the nonequilibrium thermodynamics of multi-component fluids that undergo chemical reactions. These thermodynamics can be rigorously described in the context of the GENERIC (general equation for the nonequilibrium reversible-irreversible coupling) framework. More specifically, this approach was recently applied to non-ideal polymer solutions [T. Indei and J. D. Schieber, J. Chem. Phys. 146, 184902 (2017)]. One of the species of the solution is an unreactive polymer chain represented by the bead-spring model. Using this detailed description as inspiration, we then make several simplifying assumptions to obtain a mean-field model for a Janus microswimmer. The swimmer model considered here consists of a polymer dumbbell in a sea of reactants. One of the beads of the dumbbell is allowed to act as a catalyst for a chemical reaction between the reactants. We show that the mean-squared displacement (MSD) of the center of mass of this Janus dumbbell exhibits ballistic behavior at time scales at which the concentration of the reactant is large. The time scales at which the ballistic behavior is observed in the MSD coincide with the time scales at which the cross-correlation between the swimmer's orientation and the direction of its displacement exhibits a maximum. Since the swimmer model was inspired by the GENERIC framework, it is possible to ensure that the entropy generation is always positive, and therefore, the second law of thermodynamics is obeyed.

Theoretical Analysis of Critical Flowable Physical Gel Cross-Linked by Metal Ions and Polyacrylamide-Derivative Associating Polymers Containing Imidazole Groups.

When the polymer chains are cross-linked by physical bonds having a finite lifetime, the relaxation time and viscosity do not diverge at the gel point though percolation occurs. These undivergent quantities are related to the finite-sized "largest relaxed cluster," which can relax before it breaks. Its size is the key rheological parameter characterizing of the critical physical gels. In order to evaluate this characteristic size, we propose here a generalized phenomenological model for the viscoelasticity of critical physical gels. We apply the theory to the previously reported experimental result for the physical gel consisting of polyacrylamide-derivative associating polymers containing imidazole groups cross-linked by coordination bonds with Ni ions. We successfully estimate the size of the largest relaxed cluster and the fractal dimension. The size is in good agreement with that estimated from the mean-square displacement of probe particles at the gel point by microrheological measurement. We also compare this system with the poly(vinyl alcohol) hydrogel cross-linked by borate ions, and show that the difference in the cluster structures is originating from the differences of precursor chain properties such as overlap concentration and radius of gyration and of the cross-linking states in these systems.

Determination of linear viscoelastic properties of an entangled polymer melt by probe rheology simulations.

Particle rheology is used to extract the linear viscoelastic properties of an entangled polymer melt from molecular dynamics simulations. The motion of a stiff, approximately spherical particle is tracked in both passive and active modes. We demonstrate that the dynamic modulus of the melt can be extracted under certain limitations using this technique. As shown before for unentangled chains [Karim et al., Phys. Rev. E 86, 051501 (2012)PLEEE81539-375510.1103/PhysRevE.86.051501], the frequency range of applicability is substantially expanded when both particle and medium inertia are properly accounted for by using our inertial version of the generalized Stokes-Einstein relation (IGSER). The system used here introduces an entanglement length d_{T}, in addition to those length scales already relevant: monomer bead size d, probe size R, polymer radius of gyration R_{g}, simulation box size L, shear wave penetration length Δ, and wave period Λ. Previously, we demonstrated a number of restrictions necessary to obtain the relevant fluid properties: continuum approximation breaks down when d≳Λ; medium inertia is important and IGSER is required when R≳Λ; and the probe should not experience hydrodynamic interaction with its periodic images, L≳Δ. These restrictions are also observed here. A simple scaling argument for entangled polymers shows that the simulation box size must scale with polymer molecular weight as M_{w}^{3}. Continuum analysis requires the existence of an added mass to the probe particle from the entrained medium but was not observed in the earlier work for unentangled chains. We confirm here that this added mass is necessary only when the thickness L_{S} of the shell around the particle that contains the added mass, L_{S}>d. We also demonstrate that the IGSER can be used to predict particle displacement over a given timescale from knowledge of medium viscoelasticity; such ability will be of interest for designing nanoparticle-based drug delivery.

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].

Determination of viscoelastic properties by analysis of probe-particle motion in molecular simulations.

We present a technique for the determination of viscoelastic properties of a medium by tracking the motion of an embedded probe particle by using molecular dynamics simulations. The approach involves the analysis of the simulated particle motion by continuum theory; it is shown to work in both passive and active modes. We demonstrate that, for passive rheology, an analysis based on the generalized Stokes-Einstein relationship is not adequate to obtain the values of the viscoelastic moduli over the frequency range studied. For both passive and active modes, it is necessary to account for the medium and particle inertia when analyzing the particle motion. For a polymer melt system consisting of short chains, the values calculated from the proposed approach are in good quantitative agreement with previous literature results that were obtained using completely different simulation approaches. The proposed particle rheology simulation technique is general and could provide insight into the characterization of the mechanical properties in biological systems, such as cellular environments and polymeric systems, such as thin films and nanocomposites that exhibit spatial variation in properties over the nanoscale.

Competing effects of particle and medium inertia on particle diffusion in viscoelastic materials, and their ramifications for passive microrheology.

We analyze the appropriate form for the generalized Stokes-Einstein relation (GSER) for viscoelastic solids and fluids when bead inertia and medium inertia are taken into account, which we call the inertial GSER. It was previously shown for Maxwell fluids that the Basset (or Boussinesq) force arising from medium inertia can act purely dissipatively at high frequencies, where elasticity of the medium is dominant. In order to elucidate the cause of this counterintuitive result, we consider Brownian motion in a purely elastic solid where ordinary Stokes-type dissipation is not possible. The fluctuation-dissipation theorem requires the presence of a dissipative mechanism for the particle to experience fluctuating Brownian forces in a purely elastic solid. We show that the mechanism for such dissipation arises from the radiation of elastic waves toward the system boundaries. The frictional force associated with this mechanism is the Basset force, and it exists only when medium inertia is taken into consideration in the analysis of such a system. We consider first a one-dimensional harmonic lattice where all terms in the generalized Langevin equation--i.e., the elastic term, the memory kernel, and Brownian forces-can be found analytically from projection-operator methods. We show that the dissipation is purely from radiation of elastic waves. A similar analysis is made on a particle in a continuum, three-dimensional purely elastic solid, where the memory kernel is determined from continuum mechanics. Again, dissipation arises only from radiation of elastic shear waves toward infinite boundaries when medium inertia is taken into account. If the medium is a viscoelastic solid, Stokes-type dissipation is possible in addition to radiational dissipation so that the wave decays at the penetration depth. Inertial motion of the bead couples with the elasticity of the viscoelastic material, resulting in a possible resonant oscillation of the mean-square displacement (MSD) of the bead. On the other hand, medium inertia (the Basset force) tends to attenuate the oscillations by the dissipation mechanism described above. Thus competition between bead inertia and medium inertia determines whether or not the MSD oscillates. We find that, if the medium density is larger than 4/7 of the bead density, the Basset damping will suppress oscillations in the MSD; this criterion is sufficient but not necessary to present oscillations.

Treating inertia in passive microbead rheology.

The dynamic modulus G(*) of a viscoelastic medium is often measured by following the trajectory of a small bead subject to Brownian motion in a method called "passive microbead rheology." This equivalence between the positional autocorrelation function of the tracer bead and G(*) is assumed via the generalized Stokes-Einstein relation (GSER). However, inertia of both bead and medium are neglected in the GSER so that the analysis based on the GSER is not valid at high frequency where inertia is important. In this paper we show how to treat both contributions to inertia properly in one-bead passive microrheological analysis. A Maxwell fluid is studied as the simplest example of a viscoelastic fluid to resolve some apparent paradoxes of eliminating inertia. In the original GSER, the mean-square displacement (MSD) of the tracer bead does not satisfy the correct initial condition. If bead inertia is considered, the proper initial condition is realized, thereby indicating an importance of including inertia, but the MSD oscillates at a time regime smaller than the relaxation time of the fluid. This behavior is rather different from the original result of the GSER and what is observed. What is more, the discrepancy from the GSER result becomes worse with decreasing bead mass, and there is an anomalous gap between the MSD derived by naïvely taking the zero-mass limit in the equation of motion and the MSD for finite bead mass as indicated by McKinley et al. [J. Rheol. 53, 1487 (2009)]. In this paper we show what is necessary to take the zero-mass limit of the bead safely and correctly without causing either the inertial oscillation or the anomalous gap, while obtaining the proper initial condition. The presence of a very small purely viscous element can be used to eliminate bead inertia safely once included in the GSER. We also show that if the medium contains relaxation times outside the window where the single-mode Maxwell behavior is observed, the oscillation can be attenuated inside the window. This attenuation is realized even in the absence of a purely viscous element. Finally, fluid inertia also affects the bead autocorrelation through the Basset force and the fluid dragged around with the bead. We show that the Basset force plays the same role as the purely viscous element in high-frequency regime, and the oscillation of MSD is suppressed if fluid density and bead density are comparable.

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

We have developed a single-chain theory that describes dynamics of associating polymer chains carrying multiple associative groups (or stickers) in the transient network formed by themselves and studied linear viscoelastic properties of this network. It is shown that if the average number N of stickers associated with the network junction per chain is large, the terminal relaxation time τ(A) that is proportional to τ(X)N(2) appears. The time τ(X) is the interval during which an associated sticker goes back to its equilibrium position by one or more dissociation steps. In this lower frequency regime ω<1/τ(X), the moduli are well described in terms of the Rouse model with the longest relaxation time τ(A). The large value of N is realized for chains carrying many stickers whose rate of association with the network junction is much larger than the dissociation rate. This associative Rouse behavior stems from the association/dissociation processes of stickers and is different from the ordinary Rouse behavior in the higher frequency regime, which is originated from the thermal segmental motion between stickers. If N is not large, the dynamic shear moduli are well described in terms of the Maxwell model characterized by a single relaxation time τ(X) in the moderate and lower frequency regimes. Thus, the transition occurs in the viscoelastic relaxation behavior from the Maxwell-type to the Rouse-type in ω<1/τ(X) as N increases. All these results are obtained under the affine deformation assumption for junction points. We also studied the effect of the junction fluctuations from the affine motion on the plateau modulus by introducing the virtual spring for bound stickers. It is shown that the plateau modulus is not affected by the junction fluctuations.

Rheological study of transient networks with junctions of limited multiplicity.

We theoretically study the viscoelastic and thermodynamic properties of transient gels comprised of telechelic associating polymers. We extend classical theories of transient networks so that correlations among polymer chains through the network junctions are taken into account. This extension enables us to investigate how rheological quantities such as elastic modulus, viscosity, and relaxation time are affected by the association equilibrium, and how these quantities are related to the aggregation number (or multiplicity) of the junctions. In this paper, we assume, in the conventional manner, that chains are elastically effective if both their ends are connected with other chains. It is shown that the dynamic shear moduli are well described in terms of the Maxwell model. As a result of the correlation, the reduced moduli (moduli divided by the polymer concentration) increase with the concentration, but become independent of the concentration in the high-concentration range. The fraction of pairwise junctions is larger at lower concentrations, indicating the presence of concatenated chains in the system, which decreases as the concentration increases. This leads to a network relaxation time that increases with the concentration.

Rheological study of transient networks with junctions of limited multiplicity. II. Sol/gel transition and rheology.

Viscoelastic and thermodynamic properties of transient gels formed by telechelic associating polymers are studied on the basis of the transient network theory that considers the correlation among polymer chains via network junctions. The global information of the gel is incorporated into the theory by introducing elastically effective chains defined according to the criterion of Scanlan [J. Polym. Sci. 43, 501 (1960)] and Case [J. Polym. Sci. 45, 397 (1960)]. We also consider the effects of superbridges whose backbone is formed by several chains connected in series and containing several breakable junctions. The dynamic shear moduli of this system are well described in terms of the Maxwell model characterized by a single relaxation time and high-frequency plateau modulus. Near the critical concentration at the sol/gel transition, superbridges become infinitely long along the backbone, thereby leading to a short relaxation time tau for the network. It is shown that tau is proportional to the concentration deviation Delta near the gelation point. The plateau modulus G(infinity) increases as the cube of Delta near the gelation point as a result of the mean-field treatment, and hence the zero-shear viscosity increases as eta(0) approximately G(infinity)tau approximately Delta(4). The present model can explain the concentration dependence of the dynamic moduli observed for aqueous solutions of telechelic poly(ethylene oxide).