Doi-Edwards Research Articles

Microrheological Approach for Probing the Entanglement Properties of Polyelectrolyte Solutions.

The entanglement dynamics and viscoelasticity of polyelectrolyte solutions remain active research topics. Previous studies have reported conflicting experimental results when compared to Dobrynin's scaling predictions derived from the Doi-Edwards (DE) tube model for entangled polymers. Herein, by combining classical bulk shear rheometry with diffusing wave spectroscopy (DWS) microrheometry, we investigate how the key viscoelastic parameters (the specific viscosity ηsp, the plateau modulus Ge, and the ratio of the reptation time to the Rouse time of an entanglement strand τrep/τe) depend on the polymer concentration for semidilute entangled (SE) solutions containing poly(sodium styrenesulfonate) with high molecular weight. Our experimental measurements yield Ge ∝ c1.51±0.04, in good agreement with the scaling of Ge ∝ c1.5 predicted by Dobrynin's model for salt-free polyelectrolyte SE solutions, suggesting that the electrostatic interaction influences the viscoelastic properties of polyelectrolyte SE solutions. On the other hand, the deviation in the scaling exponent for ηsp ∝ c2.56±0.04 and τrep/τe ∝ c1.82±0.28 is observed between our DWS experiments and Dobrynin's model prediction (∝ c1.5), likely due to the fact that Dobrynin's scaling model does not account for mechanisms such as the contour length fluctuation, the constraint release, and the retardation of solvent dynamics, which are known to occur for SE solutions of neutral polymers. Our results demonstrate that DWS serves as a powerful rheological tool to study the entanglement dynamics of polyelectrolyte solutions. The scaling relationships obtained in this study provide new insights to the long-standing debate on the entanglement dynamics of polyelectrolyte solutions.

Modified scaling principle for rotational relaxation in a model for suspensions of rigid rods

We have performed simulations of the model of infinitely thin rigid rods undergoing rotational and translational diffusion, subject to the restriction that no two rods can cross one another, for various concentrations well into the semidilute regime. We used a modification of the algorithm of Doi et al. [J. Phys. Soc. Jpn. 53, 3000 (1984)] that simulates diffusive dynamics using a Monte Carlo method and a nonzero time step. In the limit of zero time step, this algorithm is an exact description of diffusive dynamics subject to the noncrossing restriction. For a wide range of concentrations in the semidilute regime, we report values of the long time rotational diffusion constant of the rods, extrapolated to the limit of zero time step, for various sets of values of the infinite dilution (bare) diffusion constants. These results are compared with the results of a previous simulation of the model by Doi et al. and of previous simulations of rods with finite aspect ratio by Fixman and by Cobb and Butler that had been extrapolated to the limit of infinitely thin rods. The predictions of the Doi-Edwards (DE) scaling law do not hold for this model for the concentrations studied. The simulation data for the model display two deviations from the predictions of the DE theory that have been observed in experimental systems in the semidilute regime, namely, the very slow approach toward DE scaling behavior as the concentration is increased and the large value of the prefactor in the DE scaling law. We present a modified scaling principle for this model that is consistent with the simulation results for a broad range of concentrations in the semidilute regime. The modified scaling principle takes into account two physical effects, which we call "leakage" and "drift," that were found to be important for the transport properties of a simpler model of nonrotating rods on a lattice [Y.-L. S. Tse and H. C. Andersen, J. Chem. Phys. 136, 024904 (2012)].

Nonlinear Stress Relaxation of Scarcely Entangled Chains in Primitive Chain Network Simulations

Polymeric liquids exhibit a crossover from the nonentangled to entangled behavior on an increase of the molecular weight M. For example, it has been established that the zero-shear viscosity h0(M ) of non-entangled low-M chains is proportional to M as described by Rouse model, whereas h0(M ) of well-entangled high-M chains is proportional to M a with a = ~ 3.4. 2) Not only for the linear viscoelastic property, the crossover has been observed also for nonlinear viscoelasticity, specifically in the shear damping function h(g), where g is a magnitude of step shear strain. h(g ) is insensitive to M and close to the Doi-Edwards (DE) prediction for high-M chains. This universal damping behavior was classified by Osaki as the type-A behavior. In contrast, the g dependence of h(g) considerably weakens with decreasing M when M is below a critical value ~ 2Me (with Me being the entanglement molecular weight), as reported by Takatori et al. and Inoue et al. for polystyrene (PS) solutions and by Vega et al. for polystyrene (PS) melts. This weak damping was classified by Osaki as the type-B behavior. A couple of theoretical attempts has been made for explaining the origin of type-B damping behavior. Kapnistos et al. have reported a weak damping of h(g ) for scarcely entangled polymer deduced from their slip-spring model where a Rouse chain is confined in a tube shaped parabolic potential. The confinement is realized by some springs having one end anchored in space and the other end connected to the chain. The spring end connected to the chain slides along the chain backbone to mimic the entanglement. Kapnistos et al. reported that if the number of slip-springs on the chain, which reflects the number of entanglement per chain, is decreased, the strain dependence of h(g) is weakened. Since there is no damping in the limit of free Rouse chain, it is rather natural that the strain dependence of h(g ) becomes weaker when the number of springs on the chain becomes smaller. They discussed that their simulation results are consistent with those for comb polymers with a small number of backbonebackbone entanglements. Another explanation of the type-B behavior was given by Kharlamov et al. They derived the distribution function of the end-to-end vector of a whole chain Ree after the chain contraction along the primitive path, and then calculated the stress from the second moment of Nonlinear Stress Relaxation of Scarcely Entangled Chains in Primitive Chain Network Simulations

A lattice model of the translational dynamics of nonrotating rigid rods

We present a lattice model of oriented, nonrotating, rigid rods in three dimensions with random walk dynamics, computer simulation results for the model, and a theory for the translational diffusion constant of the rods in the perpendicular direction, D(⊥), in the semidilute regime. The theory is based on the "tube model" of Doi-Edwards (DE) theory for the rotational diffusion constant of rods that can both translate and rotate in continuous space. The theory predicts that D(⊥) is proportional to (νL(3))(-2), where ν is the concentration of rods and L is the length of the rods, which is analogous to the Doi-Edwards scaling law for rotational diffusion. The simulations find that, as νL(3) is increased, the approach to the limit of DE scaling is slow, and the -2 power in the DE scaling law is never quite achieved even at the highest concentration (νL(3) = 200) simulated. We formulate a quantitative theory for the prefactor in the scaling relationship using only DE ideas, but it predicts a proportionality constant that is much too small. To explain this discrepancy, we modify the DE approach to obtain a more accurate estimate of the average tube radius and take into account effects of perpendicular motion of rods that are not included in the original DE theory. With these changes, the theory predicts values of D(⊥) that are in much better agreement with the simulations. We propose a new scaling relationship that fits the data very well. This relationship suggests that the DE scaling law is the correct description of the scaling for infinitely thin rods only in the limit of infinite concentration, and that corrections to the DE scaling law because of finite concentration are significant even at concentrations that are well inside the semidilute regime. The implications of these results for the DE theory of rotating rods are discussed.

Reversed planar elongation of soft polymeric networks

The newly developed planar elongation fixture, designed as an add-on to the filament stretch rheometer, is used to measure reversible large amplitude planar elongation on soft elastomers. The concept of this new fixture is to elongate an annulus, by keeping the perimeter constant. The deformation on the cylindrical probe is measured using digital imaging, and it is found that the diameter only changes a few percent compared with the initial diameter. Additionally, it is found that a time-strain separable K-BKZ model based on a Doi–Edwards (DE) network strain (without independent alignment) captures the experimental data well. In particular, it is observed that it reproduces the deformation on the cylindrical probe, and this observation confirms our previous statement that the deformation on the cylindrical probe is highly sensitive towards the choice of strain tensor. The stress itself is well described by both the modified Lodge and the DE network strain. The amount of work needed during the deformation is calculated, and it is illustrated how the sample itself contributes with work upon flow reversal. As the stress is time-strain separable the energy loss originates from linear viscoelastic relaxation even in the nonlinear regime, and the loss can be determined solely from linear viscoelastic measurements.

Slip-link simulations of entangled polymers in planar extensional flow: Disentanglement modified extensional thinning

In the present work, we have extended a slip-link based model, originally proposed by Masubuchi et al. [J. Chem. Phys. 115, 4387 (2001)], to simulate planar extensional flow. The salient features of the extended model are (i) an unrestricted simulation time for planar extensional flow using Kraynik–Reinelt periodic boundary conditions, (ii) an implicit time stepping scheme to enhance numerical stability, and (iii) a mesh-free Lagrangian method for the evaluation of the osmotic force. With the new code, we have simulated both the static and dynamic responses of monodisperse linear worm-like and inverse Langevin chains and compared the simulations results with the experimental data reported by Bach et al. [J. Rheol. 47, 429 (2003)]. The simulations were run for monodisperse linear chains including up to 50 entanglements. The model is able to reasonably capture the rheological behavior of entangled systems in the nonlinear range. Based on our simulations, we also present an alternative explanation for the dynamics behind extensional thinning based on disentanglement and a physical explanation for the failure of standard models, i.e., Doi-Edwards (DE)/Doi-Edwards-Marrucci-Grizzuti (DEMG) (which predict a thinning exponent of −1) in predicting the extensional thinning exponent of near −0.5 (as seen in various experiments [Bach et al., J. Rheol. 47, 429 (2003); Luap et al., Rheol. Acta 45, 83 (2005)]).

Thermodynamic admissibility of the reptation model

In a recent paper, Ottinger and Beris assert that the celebrated Doi–Edwards (DE) model for entangled polymer dynamics is incompatible with the so-called “GENERIC” form of nonequilibrium thermodynamics. Simply put, they claim that the DE model for tube dynamics is inconsistent with the principle of virtual work. I show that with the correct choice of effective Hamiltonian, namely the full tube configurational entropy, the DE stress tensor does in fact result from a virtual work argument.

Significance of the Longest Rouse Relaxation Time in the Stress Relaxation Process at Large Deformation of Entangled Polymer Solutions

The significance of linear density equilibration time, 2τR, of a polymer chain in Doi−Edwards (DE) tube model theory was investigated for the relaxation modulus, G(t,γ), at various magnitudes of sh...

Numerical studies of viscoelastic flows using a model for entangled polymer solutions with a shear stress maximum

This paper examines the predicted flow behavior of a recently published constitutive model (the RHL model) for entangled polymeric fluids in a co-rotating two-roll mill. The constitutive model is an approximation of the Doi–Edwards–Marrucci–Grizzuti (DEMG) model. The DEMG model incorporates an explicit description of chain stretching into the basic Doi–Edwards (DE) theory. Thus, the DEMG and the approximate RHL models both exhibit strain-rate softening (or thinning) behavior similar to the DE model for strain rates that are less than the inverse of the largest Rouse relaxation time. This behavior includes the well-known but controversial prediction of a non-monotonic dependence of the shear stress on the shear rate in steady, viscometric flows. The objectives of the present work are two-fold. First, we explore whether the flow simulations capture the decrease in the strength of the extensional flow at the stagnation point of the two-roll mill that is observed experimentally for entangled polymer solutions. A similar decrease in the extensional strength of the flow is both observed and predicted for dilute polymer solutions, but for these systems this behavior is a consequence of the fact that the extensional viscosity increases monotonically with increasing shear rate. On the other hand, for entangled polymer solutions, as well as the RHL approximation to the DE theory, the viscosity decreases with increasing strain rate for strain rates smaller than the inverse of the longest Rouse relaxation time. The second objective is to explore how the non-monotonicity of the stress is manifested in a complex flow system where the flow-type is generally not close to viscometric, and the shear rate varies with position along streamlines in regions where the flow is nearly viscometric. In order to understand the signatures of the shear stress maximum for the complex flow in the co-rotating two-roll mill, we also briefly consider the numerically predicted behavior for two-dimensional channel flow and for Couette flow, both of which have been studied earlier by other authors using different models with a non-monotonic shear stress.

A Molecular Theory for Fast Flows of Entangled Polymers

The Doi−Edwards (DE) theory for the rheological properties of entangled polymer melts and solutions successfully predicts the response to large step-shear strains but fails to predict other nonlinear shear properties, such as the steady-state viscosity or the relaxation of stress after cessation of steady shearing. Many of these failures remain even in the extension of the theory by Marrucci and Grizzuti (Gazz. Chim. Ital. 1988, 118, 179)1 to allow deformation-induced “tube stretch”. Here, we find that a much more successful theory can be obtained by also accounting for “convective constraint release”, i.e., the loss of entanglement constraints caused by the retraction of surrounding chains in their tubes. (Marrucci, G. J. Non-Newtonian Fluid Mech. 1996, 62, 279 and Ianniruberto, G.; Marrucci, G. J. Non-Newtonian Fluid Mech. 1996, 65, 241).2,3 In the molecular model developed here, convective constraint release can both shorten the reptation tube and allow reorientation of interior tube segments. The revised model predicts many of the features of steady and transient shearing flows. These include a region of nearly constant steady-state shear stress at shear rates between the inverse zero-shear reptation time and the inverse Rouse time, similar to that seen in the experiments of Bercea et al. (Macromolecules 1993, 26, 7095)4 and also predicted by Marrucci and Ianniruberto (Macromol. Symp. 1997, 117, 233).5 The predictions of transient stresses after startup and cessation of shear are also in good agreement with experiments, as are predictions of nonmonotonicity in the extinction angle after stepup or stepdown in shear rate.

The effect of concentration, temperature, and molecular weight on the dynamics of rigid-rod molecules in semi-dilute solutions

The dynamics of rigid-rod-like molecules are studied using rheo-optical techniques. Measurements of flow birefringence as a function of shear rate are utilized to understand the scaling behavior of rotational diffusivity with respect to concentration and temperature. The concentration scaling exponent increases with increasing concentration and the scaling laws are valid in narrow concentration windows. The Doi-Edwards (DE) scaling law Dr ∼ c−2, holds at very high concentrations (cL3 > 150). The concentration scaling exponent decreases dramatically with increasing temperature at concentrations, cL2d > 1. Scaling of rotational diffusivity, with respect to temperature and solvent viscosity in the semidilute regime, does not follow the predictions of DE theory (and related caging ideas). On the contrary, a model proposed by Fixman was found to explain both the temperature and concentration dependence of the rotational diffusivity. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 181–190, 1998

First and second normal stress difference relaxation in reversing double‐step strain flows

Reversing double‐step strains provide a severe test of constitutive equations and have often been used to test the Doi–Edwards (DE) model with and without the independent alignment approximation. We report measurements of the full stress tensor in a concentrated monodisperse polystyrene solution subjected to reversing double‐step strains using flow birefringence. Shear stress and first normal stress difference results agree with previous studies. In flows where the second strain is half the magnitude of the first (‘‘specialized type‐B’’), certain rheological models predict that normal stresses should be independent of time between strains (t1), and equal to those measured in single‐step strain. Both first and second normal stress differences follow this behavior at long times, but deviations are found at short times, where chain retraction is occurring. In flows with equal and opposite step strains (‘‘type‐C’’), the ratio N1/σ is found to be equal to the strain. In both of these flows, the DE model predicts the normal stress ratio −N2/N1 to be independent of t1. In specialized type‐B flows, the experimental normal stress ratio is nearly independent of t1, while in type‐C flows the ratio depends strongly on t1 and is found to be substantially larger than that predicted by the DE model or observed in single‐step strains. DE calculations for more general flow protocols predict that the normal stress ratio may depend on t1 and on the time following the second deformation. Experiments show qualitative agreement with both forms of the DE model for these other flows, where the major influence of the independent alignment approximation is on the magnitude of the predicted normal stress ratio.

Doi‐Edwards theory evaluation in double‐step strain flows

AbstractIt is widely accepted that for reversing double‐step strain deformations, predictions based on the Doi‐Edwards (DE) molecular theory without the independent alignment approximation (IAA) are superior to predictions obtained with the IAA, or equivalently, the Kaye‐Bernstein‐Kearsley‐Zapas (K‐BKZ) theory. This summation, however, is based on data obtained over limited ranges of strain and time: the time both between the step strains (t1) and following the second step strain (t–t1). In this study, a thorough evaluation of the DE theory is carried out using a comprehensive double‐step strain flow data set. The results of this study indicate that the DE theory is an improvement over the K‐BKZ theory in flows with strain reversal but only for cases when the criteria t1, t–t1 ≫ τk is satisfied. The constant τk, defined as the time beyond which the stress relaxation modulus is factorable: G(γ, t) = h(γ)G(t), is believed to represent the end of the chain retraction process in the DE theory. It appears that the dynamics of chain retraction have an important influence on double‐step strain behavior and, therefore, should be accounted for in molecular‐based theories devised to have general validity in this important deformation history. © 1994 John Wiley & Sons, Inc.

Flexural perturbations of free jets of maxwell and Doi-Edwards liquids

Flexural perturbations of high-velocity free jets of drop liquids moving in air are reinforced by the fact that the air pressure on the concave sections of the jet surface is greater than on the convex sections. The linear and nonlinear stages of development of flexural perturbations were studied in [1–5] for viscous Newtonian fluids. The effect of elastic stresses in the fluid on the growth of flexural perturbations of jets was first examined in [6], where it was assumed in an analysis of the growth of small disturbances that surface tension was constant along the jet, i.e., the investigators actually studied a tensed string. The studies [7, 8] examined the linear stage of growth of flexural perturbations of jets of Maxwell liquids. Our goal here is to analyze the dynamics of long-wave flexural perturbations of jets of viscoelastic fluids in both the linear and nonlinear stages of development. The rheological behavior of the fluid is described by two models — the phenomenological (Maxwell) model and the physical-molecular (Doi-Edwards) model. It is shown that the disturbances are oscillatory in character in the nonlinear stage of development. Meanwhile, the results of calculations performed with the Maxwell (M) and Doi-Edwards (DE) rheological models in the given problem agree with each other quantitatively as well as qualitatively.

Chain retraction, length fluctuations, and small‐angle neutron scattering in strained polymer melts

AbstractWe propose a possible explanation for an apparent contradiction between the Doi–Edwards (DE) theory and time‐resolved small‐angle neutron scattering experiments. The original DE theory predicts that a chain in a melt subjected to a step‐elongation undergoes a reduction of its dimensions in both the parallel and perpendicular directions to the strain axis, before returning to an isotropic conformation by reptation. In the present paper, we propose a crude model to compare the relative effect of retraction and of the chain length fluctuations, introduced later by Doi to explain the M power law of viscosity of polymer melts. We show that length fluctuations are able to screen retraction in most experimental situations available to small‐angle neutron scattering, whereas that is not the case for viscoelasticity. Quantitative predictions are consistent with the experimental data presently available.