Finite Extensibility Effects Research Articles

Modeling microgel swelling: Influence of chain finite extensibility.

Microgels exhibit the ability to undergo reversible swelling in response to shifts in environmental factors that include variations in temperature, concentration, and pH. While several models have been put forward to elucidate specific aspects of microgel swelling and its impact on bulk behavior, a consistent theoretical description that chains throughout the microscopic degrees of freedom with suspension properties and deepens into the full implications of swelling remains a challenge yet to be met. In this work, we extend the mean-field swelling model of microgels from Denton and Tang [J. Chem. Phys. 145, 164901 (2016)] to include the finite extensibility of the polymer chains. The elastic contribution to swelling in the original work is formulated for Gaussian chains. By using the Langevin chain model, we modify this elastic contribution in order to account for finite extensibility effects, which become prominent for microgels containing highly charged polyelectrolytes and short polymer chains. We assess the performance of both elastic models, namely for Gaussian and Langevin chains, comparing against coarse-grained bead-spring simulations of ionic microgels with explicit electrostatic interactions. We examine the applicability scope of the models under a variation of parameters, such as ionization degree, microgel concentration, and salt concentration. The models are also tested against experimental results. This work broadens the applicability of the microgel swelling model toward a more realistic description, which brings advantages when describing the suspensions of nanogels and weak-polyelectrolyte micro-/nanogels.

Master curves for FENE-P fluids in steady shear flow

The FENE-P (Finitely-Extensible Nonlinear Elastic) dumbbell constitutive equation is widely used in simulations and stability analyses of free and wall-bounded viscoelastic shear flows due to its relative simplicity and accuracy in predicting macroscopic properties of dilute polymer solutions. The model contains three independent material parameters, which expressed in dimensionless form correspond to a Weissenberg number ( Wi ), i.e. , the ratio of the dumbbell relaxation time scale to a characteristic flow time scale, a finite extensibility parameter ( L ), corresponding to the ratio of the fully extended dumbbell length to the root mean square end-to-end separation of the polymer chain under equilibrium conditions, and a solvent viscosity ratio, commonly denoted β . An exact solution for the rheological predictions of the FENE-P model in steady simple shear flow is available (Sureshkumar et al., 1997), but the resulting nonlinear and nested set of equations do not readily reveal the key shear-thinning physics that dominates at high Wi as a result of the finite extensibility of the polymer chain. In this note we review a simple way of evaluating the steady material functions characterizing the nonlinear evolution of the polymeric contributions to the shear stress and first normal stress difference as the shear rate increases, provide asymptotic expansions as a function of Wi , and show that it is in fact possible to construct universal master curves for these two material functions as well as the corresponding stress ratio. Steady shear flow experiments on three highly elastic dilute polymer solutions of different finite extensibilities also follow the identified master curves. The governing dimensionless parameter for these master curves is Wi / L and it is only in strong shear flows exceeding Wi / L ≳ 1 that the effects of finite extensibility of the polymer chains dominate the evolution of polymeric stresses in the flow field. We suggest that reporting the magnitude of Wi / L when performing stability analyses or simulating shear-dominated flows with the FENE-P model will help clarify finite extensibility effects. • Asymptotic solutions for FENE-P model in steady shear flow are calculated. • Master curves exist for material functions of FENE-P fluids in steady shear flow. • Governing dimensionless parameter is (Weissenberg number)/(finite extensibility). • Reporting this dimensionless parameter clarifies the magnitude of nonlinear effects.

Nonlinear rheology of entangled wormlike micellar solutions predicted by a micelle-slip-spring model

We examine linear and nonlinear shear and extensional rheological properties using a “micelle-slip-spring model” [T. Sato et al., J. Rheol. 64, 1045–1061 (2020)] that incorporates breakage and rejoining events into the slip-spring model originally developed by Likhtman [Macromolecules 38, 6128–6139 (2005)] for unbreakable polymers. We here employ the Fraenkel potential for main chain springs and slip-springs to address the effect of finite extensibility. Moreover, to improve extensional properties under a strong extensional flow, stress-induced micelle breakage (SIMB) is incorporated into the micelle-slip-spring model. Thus, this model is the first model that includes the entanglement constraint, Rouse modes, finite extensibility, breakage and rejoining events, and stress-induced micelle breakage. Computational expense currently limits the model to micellar solutions with moderate numbers of entanglements (≲7), but for such solutions, nearly quantitative agreement is attained for the start-up of the shearing flow. The model in the extensional flow cannot yet be tested owing to the lack of data for this entanglement level. The transient and steady shear properties predicted by the micelle-slip-spring model for a moderate shear rate region without significant chain stretch are fit well by the Giesekus model but not by the Phan–Thien/Tanner (PTT) model, which is consistent with the ability of the Giesekus model to match experimental shear data. The extensional viscosities obtained by the micelle-slip-spring model with SIMB show thickening followed by thinning, which is in qualitative agreement with experimental trends. Additionally, the extensional rheological properties of the micelle-slip-spring model with or without SIMB are poorly predicted by both the Giesekus and the PTT models using a single nonlinear parameter. Thus, future work should seek a constitutive model able to capture the behavior of the slip-spring model in shear and extensional flows and so provide an accurate, efficient model of micellar solution rheology.

Stress in a polymer brush

We study the stress distribution in a polymer brush material over a range of graft densities, using theory and molecular dynamics (MD) simulations. MD simulations confirmed the quartic variation of the normal stress within the bulk of the brush, as predicted in our previous work. However, in the high graft density regime, further improvements to the theory are needed, as it is known that finite extensibility effects (force-extension divergence) invalidate Gaussian elasticity of chains, and the restriction to binary interaction among monomers is insufficient. This motivated us to extend a semi-analytical strong stretching theory (SST) for polymer brushes that use Langevin force-extension relation for chains and a modified Carnahan–Starling (CS) equation of state to model monomer interactions. Our extended theory elucidates the stress profiles obtained from MD simulations. It reproduces Gaussian chain results for small graft densities, and shows a good qualitative agreement with MD results for monomer density profile, end density profile and stress profile for high graft densities without the need for fitting parameters (virial coefficients). Quantitative comparisons of MD results with various available theories suggest that excluded volume correlations may be important.

Effect of Finite Extensibility on Nonlinear Extensional Rheology of Polymer Melts

We recently showed that two systems have the same nonlinear flow dynamics if they have (1) the same number of entanglements, Z, (2) the same number of Kuhn segments between entanglements, Ne, and (3) a single monomeric friction reduction between all species. Because differences in polymer chemistry result in different values of (1)–(3), these results show that the likelihood of two melts having the same nonlinear behavior is improbable. This work determines the dependence of the nonlinear extensional flow behavior on Ne for three polymer melts: polystyrene, poly(methyl methacrylate), and poly(tert-butylstyrene). We show that polymer melts with the same Z have almost identical scaled linear viscoelasticity. Extensional rheology at constant strain rate shows strain hardening depends strongly on the value of Ne. More specifically, our data suggest a power-law dependence of steady state stress on the Weissenberg number which increases with increasing Ne.

On the origin of brittle fracture of entangled polymer solutions and melts

A novel criterion for brittle fracture of entangled polymer liquids is presented: Crack initiation follows from rupture of primary C–C bonds, when the strain energy of an entanglement segment reaches the energy of the covalent bond. Thermal fluctuations lead to a short-time concentration of the strain energy on one C–C bond of the entanglement segment, and the chain ruptures. This limits the maximum achievable stretch of entanglement segments to a critical stretch of fc≤6. Recent experimental data of Huang et al. [Phys. Rev. Lett. 117, 087801 (2016)]] and Huang and Hassager [Soft Matter 13, 3470–3474 (2017)] on fracture of solutions of nearly monodisperse polystyrenes dissolved in oligomeric styrene and of a well characterized polydisperse polystyrene melt, are in general agreement with this fracture criterion. For quantitative agreement, finite extensibility effects have to be considered.

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].

Coil-bridge transition in a single polymer chain as an unconventional phase transition: theory and simulation.

The coil-bridge transition in a self-avoiding lattice chain with one end fixed at height H above the attractive planar surface is investigated by theory and Monte Carlo simulation. We focus on the details of the first-order phase transition between the coil state at large height H ⩾ Htr and a bridge state at H ⩽ Htr, where Htr corresponds to the coil-bridge transition point. The equilibrium properties of the chain were calculated using the Monte Carlo pruned-enriched Rosenbluth method in the moderate adsorption regime at (H/Na)tr ⩽ 0.27 where N is the number of monomer units of linear size a. An analytical theory of the coil-bridge transition for lattice chains with excluded volume interactions is presented in this regime. The theory provides an excellent quantitative description of numerical results at all heights, 10 ⩽ H/a ⩽ 320 and all chain lengths 40 < N < 2560 without free fitting parameters. A simple theory taking into account the effect of finite extensibility of the lattice chain in the strong adsorption regime at (H/Na)tr ⩾ 0.5 is presented. We discuss some unconventional properties of the coil-bridge transition: the absence of phase coexistence, two micro-phases involved in the bridge state, and abnormal behavior in the microcanonical ensemble.

Nonequilibrium molecular dynamics simulation of dendrimers and hyperbranched polymer melts undergoing planar elongational flow

The planar elongational melt rheology and structural properties of dendrimers and hyperbranched polymers of different molecular weights (generations 1–4) and their linear counterparts have been studied using nonequilibrium molecular dynamics simulation techniques in the isothermal-isobaric ensemble. The extensional viscosity showed three distinctive regions against strain-rate, including an initial Newtonian region at low strain-rates, followed by a thickening behavior at medium strain-rates and terminated with a thinning region at very high strain-rates, in agreement with the Sarkar and Gupta model [J. Rein. Plast. Comp. 20, 1473–1484 (2001)]. In addition, a structural analysis was performed to study the size, shape, and spatial distributions within globular dendrimers and hyperbranched polymer molecules under planar elongational flow (PEF). Ratios of the eigenvalues of the gyration tensor showed that contrary to shear flow, under PEF even at low strain rates, dendrimers and hyperbranched molecules have ellipsoidal conformations and change to a much more flattened prolate shape at higher strain rates. In combination with the eigenvalue ratios, the distribution of monomers from the central core of the molecules showed that the thickening region occurs due to branches being stretched, and terminal thinning behavior stems primarily from flow-induced alignment and finite extensibility effects.

Effect of Finite Extensibility on the Equilibrium Chain Size

AbstractWe investigate the finite‐extensibility effect on the equilibrium size of a single polymer chain by using a Flory‐type calculation. The finite extensibility of the chain is effectively taken into account by modifying the Gaussian stretching energy to a non‐Gaussian form which recovers the harmonic spring behavior in the limit of weak stretch and diverges when the chain extension approaches the total contour length. In general, the finite extensibility decreases the equilibrium size of the chain compared to that obtained from Flory's classic theory, due to a steeper free energy or effective potential of the chain generated by the finite‐extensibility effect. It is illustrated clearly that, for a chain with a short contour length and/or with strong repulsions between monomers, the finite‐extensibility correction is important and necessary to be taken into account. A typical system where the finite extensibility plays an important role is suggested to be the highly charged and highly crosslinked hydrogel.magnified image

Verification of branch point withdrawal in elongational flow of pom-pom polystyrene melt

According to tube model ideas, chain stretch at deformation rates below the inverse Rouse time of the chain, is only possible for polymer topologies with two or more branch points. The basic topologies, which embody this idea, are the H molecule with two side chains, and the pom-pom molecule with q&amp;gt;2 side chains at each end of the backbone. According to the pom-pom hypothesis, maximum chain stretch of the backbone is limited by branch point withdrawal, i.e., the side chains are drawn into the tube of the backbone as soon as the relative tension in the backbone reaches a value of q. This hypothesis, which has never been verified before, can now be tested by considering recent elongational experiments by Nielsen et al. [Macromolecules 39, 8844–8853 (2006)] on a nearly monodisperse polystyrene pom-pom melt with q=2.5. The analysis presented is based on the original integral version of the pom-pom model, and on the molecular stress function (MSF) model with strain-dependent tube diameter. The material strain measure determined from the experiments is found to be consistent with a constant maximum stretch, independent of the elongation rate, which is, however, significantly larger than q. To achieve quantitative agreement between experiment and modeling, (1) dynamic dilution of the backbone, which increases the tube diameter of the backbone and reduces equilibrium tension in the backbone, (2) finite extensibility effects, (3) stretch relaxation causing a transition from chain stretch to tube squeeze at lower strain rates, and (4) the dynamics of branch point withdrawal need to be considered. Integrating all of these features in a MSF stretch evolution equation with multiple time scales, the fundamental pom-pom hypothesis is confirmed.

Statics, linear, and nonlinear dynamics of entangled polystyrene melts simulated through the primitive chain network model

Simulation results of the primitive chain network (PCN) model for entangled polymers are compared here to existing data of diffusion coefficient, linear and nonlinear shear and elongational rheology of monodisperse polystyrene melts. Since the plateau modulus of polystyrene is well known from the literature, the quantitative comparison between the whole set of data and simulations only requires a single adjustable parameter, namely, a basic time. The latter, however, must be consistent with the known rheology of unentangled polystyrene melts, i.e., with Rouse behavior, and is therefore not really an adjustable parameter. The PCN model adopted here is a refined version of the original model, incorporating among other things a more accurate description of chain end dynamics as well as finite extensibility effects. In the new version, we find good agreement with linear rheology, virtually without adjustable parameters. It is also shown that, at equilibrium, Gaussian statistics are well obeyed in the simulated network. In the nonlinear range, excellent agreement with data is found in shear, whereas discrepancies and possible inadequacies of the model emerge in fast uniaxial elongational flows, even when accounting for finite extensibility of the network strands.

Modeling of the effect of rigid fillers on the stiffness of rubbers

AbstractThe theories that predict the increase in the modulus of elastomers resulting from the presence of a rigid filler are typically derived from Einstein's viscosity law. For example, Guth and Gold used this approach to predict how the Young's modulus of an elastomer is related to the filler volume fraction. Hon et al. have shown using finite element microstructural models that stiffness predictions at small strains were also possible. Here, microstructural finite element models have been used to investigate the modulus of filled elastomer over a wider range of strains than has been possible previously. At larger strains, finite extensibility effects are significant and here an appropriate stored energy function proposed by Gent was adopted. In this work, the effect of spherical MT‐type carbon‐black filler behavior was considered. Different models were made and the results were then compared to experimental measurement of the stiffness taken from the literature. It is shown that the boundary conditions of the microstructural unit cell lie between the two extremes of free surfaces and planar surfaces. Also as the filler volume fraction increases then the number of filler particles required in the representative volume to predict the actual stiffness behavior also increases. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci, 2008

Effect of purely elastic flow instability on molecular conformation and drag

Linear stability analysis has shown that viscoelastic creeping flow of an Oldroyd-B liquid through a sinusoidal channel is unstable to stationary, wall-localized and short wavelength perturbations [B. Sadanandan, R. Sureshkumar, Global linear stability analysis of non-separated viscoelastic flow through a periodically constricted channel, J. Non-Newtonian Fluid Mech. 122 (2004) 55]. In this work, time-dependent simulations are performed to determine the nonlinear evolution of finite amplitude disturbances in the post-critical flow regime. It is shown that a nonlinear transition, which is facilitated by a supercritical pitchfork bifurcation, establishes a finite amplitude state (FAS) in which the average polymer stretch is highly modulated. The maximum normal stress, observed at the channel nip, can increase by up to approximately 100% when the Weissenberg number, defined as the ratio of the fluid relaxation time to an inverse characteristic shear rate, is increased by only 10% beyond its critical value. This is attributed to the amplification of configurational perturbations by the base flow shear rate, which attains its maximum at the channel nip. The effect of finite chain extensibility on the critical condition and nonlinear instability is investigated using the FENE-CR model. The stabilizing effect of finite extensibility can be expressed through a renormalization of the Weissenberg number by accounting for the screening effect of the nonlinear force law on the transmission of configurational perturbations to polymeric stress. The principal features of the FAS are qualitatively model-independent. The FAS exhibits a small, but numerically perceptible increase in the friction factor as compared to the base flow. The implication of the findings on the experimentally observed flow resistance enhancement phenomenon in viscoelastic creeping flows through converging/diverging geometries is discussed.

Linear to Non-linear Rheology of Wheat Flour Dough

Abstract We provide an overview of transient extensional rheometry techniques for wheat flour doughs in which the deformation and material response is well defined. The behavior of a range of model doughs was explored with a Filament Stretching Extensional Rheometer (FISER). The measurements were also compared to data obtained with a new wind- up extensional rheometer; the SER universal testing platform. A simple empirical constitutive equation, which allows characterization of the experimental results with a small number of parameters, is presented to describe the resulting measurements. To characterize the relaxation modulus of the doughs, small amplitude shear tests were performed on samples that have been shear-mixed in a mixograph for varying lengths of time. The linear viscoelastic properties were found to exhibit a broad power-law dependence on the imposed oscillatory frequency that is very reminiscent of that exhibited by a critical gel. The critical gel model of Winter and Chambon [1, 2] was used as the basis for constructing a non-linear constitutive equation for the material stress by combining the relaxation modulus for the critical gel with a Lodge rubber-like liquid form for the kinematics. Transient uniaxial extensional data recorded from the FISER and SER instruments were then compared to the predictions of the constitutive equation. The model captures the initial power- law response and subsequent strain-hardening; however additional physics is required to describe the rheological phenomena at very large Hencky strains, including finite extensibility effects and filament rupture in extensional flows.

Scission and recombination efficiency of hybrid crosslinks in natural rubber

Adding a post-vulcanisation stabiliser (PVS) to a natural rubber mix forms hybrid crosslinks. One PVS is a hexamethylene-1,6-bisthiosulphate disodium salt dihydrate (HTS) that promotes the formation of flexible hybrid crosslinks. It is said that the fatigue life of natural rubber vulcanisates containing hybrid crosslinks is enhanced because of the flexible nature of such crosslinks. However, based on current literature, it is claimed that improvement in fatigue life is due to an energy dissipation process within the vulcanisate engendered by the scission and recombination of network chains. Thus, the aim of the investigation was to ascertain whether hybrid crosslinks break and recombine in the same way, as do crosslinks in natural rubber vulcanisates introduced by conventional curing systems. Results indicate that non-affine or finite extensibility effects cause scission and recombination of crosslinks. Such mechanisms manifest themselves as permanent set. At normal levels of crosslink density these effects appear to be very pronounced in vulcanisates containing HTS compared with vulcanisates containing conventional curing systems. Enhanced non-affine behaviour in vulcanisates containing HTS appears to be associated with poor dispersion. Inadequate dispersion can lead to localised areas of high crosslink density. Such areas may be regarded as being the major source contributing to finite extensibility effects. The present work has lead to an interesting paradox: inadequate dispersion in compounding is undesirable; however, inadvertently poor dispersion seems to be responsible for enhanced fatigue life, a feature that is clearly highly desirable.

First-order coil-to-flower transition of a polymer chain pinned near a stepwise external potential: Numerical, analytical, and scaling analysis

A polymer chain near a penetrable interface is studied in the Gaussian model, in the lattice random walk model and by a scaling analysis. The interface is modeled as an external potential u of a Heaviside step-function form. One end of the chain is fixed at a distance z0 away from this interface. When the end point is fixed in the high potential region, a first-order coil-to-flower transition takes place upon variation of the distance z0. Here, the flower has a strongly stretched stem from the grafting point towards the interface and, on top of it, a crown composed of the remaining segments in a (perturbed) coil conformation. The coil-to-flower transition is analyzed in terms of the Landau free energy. The order parameter is taken to be related to the fraction of segments residing in the energetically favorable region. Exact analytical expressions for the Landau function are obtained in the Gaussian model for any distances z0 and potential strength u. A phase diagram in the z0 versus u coordinates is constructed. It contains a line of the first-order phase transitions (binodal line) ending at a critical point z0=u=0, and two spinodal lines. Numerical results are obtained for several chain lengths in the lattice random walk model demonstrating the effects of finite extensibility on the position of the transition point. Excluded volume effects are analyzed within the scaling approach.

Excluded volume effects on the structure of a linear polymer under shear flow

The effect of excluded volume interactions on the structure of a polymer in shear flow is investigated by Brownian dynamics simulations for chains with size 30⩽N⩽300. The main results concern the structure factor S(q) of chains of N=300 Kuhn segments, observed at a reduced shear rate β=γ̇τ=3.2, where γ̇ is the bare shear rate and τ is the longest relaxation time of the chain. At low q, where anisotropic global deformation is probed, the chain form factor is shown to match the form factor of the continuous Rouse model under shear at the same reduced shear rate, computed here for the first time in a wide range of wave vectors. At high q, the chain structure factor evolves towards the isotropic equilibrium power law q−1/ν typical of self-avoiding walk statistics. The matching between the excluded volume and ideal chains at small q, and the excluded volume power law behavior at large q are observed for q orthogonal to the main elongation axis but not yet for q along the elongation direction itself, as a result of interferences with finite extensibility effects. Our simulations support the existence of anisotropic shear blobs for polymers in good solvent under shear flow for β&amp;gt;1 provided chains are sufficiently long.

A Rheo-Optical Study on Polystyrene under Large Tensile Deformation around the Glass Transition Temperature

The tensile stress and the birefringence of polystyrene were measured under elongation at a constant rate up to the elongation ratio of 4 at 115, 105, and 100 °C. The tensile stress was separated into two component functions, designated with subscripts R and G, through a modified stress−optical rule, MSOR, considering the effect of finite extensibility of the polymer chain on the stress dependence of the stress-optical coefficient. The R component represents the stress attributed to the polymer segment orientation and the G component to the twist of the polymer chain. At 115 °C, the time dependence of the viscosity growth function of the two components, ηER+ and ηEG+, can be described in the framework of linear viscoelasticity except for a steep increase of ηER+ at times longer than 1/(2e), where e is the rate of elongation. It is well-known that this type of steep increase is due to the strong increase of strain measured under elongation at a constant rate. At lower temperatures, ηEG+ decreased with incr...

Finite extensibility effects in nematic elastomers

We present a model calculation of the elasticity of a nematic elastomer of finite-length network strands under affine deformation. The result is derived in its full tensorial structure, and its predictions are examined. In particular, we look at the non-Gaussian corrections to the effects of soft elasticity and elastic anisotropy. We find that the soft modes can at most be hardened at O(N−2) when compared to the conventional rubber elastic energy, where N is the number of monomer units along a typical strand. The elastic moduli for extensions parallel and perpendicular to the nematic director are found to differ at O(N−1).