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We report on shear startup data for two wormlike micellar solutions, differing only in concentration and type of two binding aromatic sodium salts. The surfactant molecule is cetylpiridinium chloride at a fixed concentration (100 mM). Sodium salicylate (NaSal) and diclofenac sodium (Diclo) are used as binding salts at concentrations 68 mM NaSal and 52 mM Diclo such that both systems are fully entangled and their linear viscoelastic response is essentially identical. Both systems show the linear response typical of a wormlike micellar solution, with terminal behavior at low frequencies, a well-defined moduli crossover, and a plateau modulus. In the nonlinear regime, however, the behavior of the two systems is totally different, suggesting that the molecular structure difference of the salts and their binding activity to the surfactant molecule are both crucial to determine the fast flow behavior. The NaSal solution shows a very complex rheological response, with strain hardening and very sharp stress peaks, whereas the solution containing Diclo behaves much like ordinary linear polymers, exhibiting pronounced overshoots as well as moderate undershoots in the transient shear viscosity, before approaching the steady state. This polymerlike behavior has also been proved by successfully comparing data with predictions of a constitutive equation recently adopted for both entangled polymers and linear wormlike micelles. As far as NaSal is concerned, a phenomenological model based on rubber network theory is developed, which describes the flow singularities. A physical interpretation of the different behavior in the nonlinear regime is also suggested.

Ianniruberto and Marrucci developed a theory whereby entangled branched polymers behave like linear ones in fast elongational flows. In order to test such prediction, Huang et al. performed elongational measurements on star polymer melts, indeed revealing that, in fast flows, the elongational viscosity is insensitive to the molecular structure, provided the molecular weight of the backbone is the same. Inspired by these studies, we here report on results obtained with multi-chain slip-link simulations for symmetric and asymmetric star polymer melts, as well as calculations of the Rouse time of the examined branched structures. The simulations semi-quantitatively reproduce the experimental data if the Kuhn-segment orientation-induced reduction of friction (SORF) is accounted for. The observed insensitivity of the nonlinear elongational viscosity to the molecular structure for the same span molar mass may be due to several factors. In the symmetric case, the calculated Rouse time of the star marginally differs from that of the linear molecule, so that possible differences in the observed stress fall within the experimental uncertainty. Secondly, it is possible that the flow-induced formation of hooked star pairs makes the effective Rouse time of the aggregate even closer to that of the linear polymer because the friction center moves towards the branchpoint of the star molecule. In the asymmetric case, it is shown that the stress contributed by the short arms is negligible with respect to that of the long ones. However, such stress-reduction is balanced by a dilution effect whereby the unstretched arms reduce SORF as they decrease the Kuhn-segment order parameter of the system. As a result of that dilution, the stress contributed by the backbone is larger. The two effects compensate one another so that the overall stress is virtually the same as the other architectures., 19 pages, 6 figures

We report on the unusual response of a well-characterized entangled comb polymer in simple shear flow. The polymer with highly entangled backbone (about 16 entanglements) and on average 29 long branches (about 3 entanglements each) has been extensively characterized by interaction chromatography and its rheology carefully studied under controlled conditions using a special cone partitioned-plate geometry. We observe that the start-up shear stress becomes roughly rate independent above a certain critical rate, related to the relaxation time of the branches. Within the rate-independent region, the start-up shear stress displays a double overshoot. We interpret these observations in light of tube-based pompom dynamics. The key idea is that for sufficiently long branches the main stress overshoot, which reflects backbone stretching and orientation, is preceded by the withdrawal of branches into the backbone tube. The excellent quantitative comparison between the simulations and experiments supports the proposed mechanism of the double stress overshoot.

We here revisit existing results of Kremer-Grest molecular dynamics simulations of polymer melts that provide values of the friction coefficient ζ of the Kremer-Grest bead for the coarse-grained Fraenkel chain model, both at equilibrium, and in very fast uniaxial extensional flows. The Fraenkel chain model is chosen because it can guarantee chain inextensibility in the flow-induced fully aligned conformation. Since the value of ζaligned for the fully aligned conformation is found to be much smaller than the equilibrium value ζeq, a controversial issue of the nonlinear rheology of polymer melts is here resolved. Indeed, this finding confirms previous evidence emerging from the rheology of several polymers, where seemingly the friction coefficient decreases with increasing flow-induced molecular coalignment. Often, however, such evidence was indirect, and the friction coefficient reduction has not been accepted by a large fraction of the scientific community.

We investigate the linear rheology of ultrahigh molecular weight polyethylene (UHMWPE) solutions with the aim of determining the molecular weight distribution of the polymer. The UHMWPE is dissolved in oligo-ethylene in order to avoid issues related to unfavorable interactions with the solvent. To prepare the solutions, UHMWPE, solvent, and a fixed amount of antioxidants are mixed by means of a corotating twin-screw microcompounder. All prepared solutions are within the concentrated regime, as confirmed by the scaling laws of the main rheological parameters (plateau modulus, relaxation time, and zero-shear viscosity) with concentration. Based on the viscoelastic response of the solutions, we adopt a heuristic approach to extrapolate the linear viscoelastic behavior of the melt, according to a time-concentration superposition principle. Such a technique allows us to span many decades of angular frequency, eventually attaining the terminal relaxation regime. The latter is difficult to achieve by direct measurements in the molten state because of experimental issues such as extremely long experimental times and thermal limits. The viscoelastic spectrum of the melt is used to obtain the molecular weight distribution (MWD) according to the time-dependent diffusion/double reptation model. The MWD of UHMWPE evaluated by using this approach agrees well with data obtained from gel permeation chromatography.

Branchpoint Withdrawal (BPW) has been recognized as one of the important molecular mechanisms for the description of the dynamics of entangled branched polymers under fast flows. However, the relation to the other known molecular mechanisms has not been fully elucidated yet. In this study we performed primitive chain network (i.e., multi-chain slip-link) Brownian simulations for a melt of a well-characterized monodisperse polystyrene H-polymer, for which the linear viscoelasticity and shear viscosity growth curves at several shear rates are available in the literature. After confirming the consistency of the simulations with the rheological data, we used the simulations to analyze the molecular motion in detail. The results reveal that molecular tumbling occurs in branched polymers just as in linear ones, and that it is accelerated by BPW. Furthermore, BPW not only mitigates backbone stretch, as expected, but also arm stretch. However, because the transient startup viscosity is anyhow dominated by chain stretch dynamics rather than by molecular tumbling, our results rationalize the fact that pom–pom theories successfully ignore tumbling in shear flows., ファイル公開：2021/01/28

In some recent experiments on entangled polymers of stress growth in startup of fast shear flows an undershoot in the shear stress is observed following the overshoot, i.e., before approaching the steady state. Whereas tumbling of the entangled chain was proposed to be at its origin, here we investigate another possible cause for the stress undershoot, i.e., slippage at the interface between polymer and solid wall. To this end, we extend the primitive chain network model to include slip at the interface between entangled polymeric liquids and solid walls with grafted polymers. We determine the slip velocity at the wall, and the shear rate in the bulk, by imposing that the shear stress in the bulk polymers is equal to that resulting from the polymers grafted at the wall. After confirming that the predicted results for the steady state are reasonable, we examine the transient behavior. The simulations confirm that slippage weakens the magnitude of the stress overshoot, as reported earlier. The undershoot is also weakened, or even disappears, because of a reduced coherence in molecular tumbling. In other words, the disentanglement between grafted and bulk chains, occurring throughout the stress overshoot region, does not contribute to the stress undershoot., Comment: 38 pages and 9 figures

We here suggest a new Brownian dynamics algorithm for telechelic associating polymers with the aim of better understanding the complex rheological behavior of these systems and in particular of hydrophobically modified ethoxylated urethane solutions showing strain hardening in shear startup and significant deviations from the Cox–Merz rule (including shear thickening phenomena). Several molecular mechanisms have indeed been suggested to explain such a complex behavior, ranging from finite extensible nonlinear elasticity (FENE) effects, shear-induced increased association, and, very recently, flow-induced repulsive interactions between colliding flower-like micelles. The preliminary results here reported (only exploring a limited range of the wide parameter space) appear to suggest that FENE effects are mostly responsible for strain hardening, while deviations from the Cox–Merz rule are due to the persistence of bridge chains, as well as, in some cases, to the flow-induced formation of new bridges. In the explored parameter space, no significant role appears to be played by flow-induced collisions between micelles beyond that of preventing network collapse.We here suggest a new Brownian dynamics algorithm for telechelic associating polymers with the aim of better understanding the complex rheological behavior of these systems and in particular of hydrophobically modified ethoxylated urethane solutions showing strain hardening in shear startup and significant deviations from the Cox–Merz rule (including shear thickening phenomena). Several molecular mechanisms have indeed been suggested to explain such a complex behavior, ranging from finite extensible nonlinear elasticity (FENE) effects, shear-induced increased association, and, very recently, flow-induced repulsive interactions between colliding flower-like micelles. The preliminary results here reported (only exploring a limited range of the wide parameter space) appear to suggest that FENE effects are mostly responsible for strain hardening, while deviations from the Cox–Merz rule are due to the persistence of bridge chains, as well as, in some cases, to the flow-induced formation of new bridges. In the ...

We report shear startup data on solutions of entangled linear wormlike micelles, only differing in the concentration of a binding aromatic salt. Wormlike micelles behave similarly to ordinary polymers in fast shear flows, exhibiting pronounced overshoots as well as tiny undershoots in transient shear viscosity, before approaching the steady state. The analogy is here emphasized by successfully comparing data with predictions of a constitutive equation recently adopted for ordinary entangled polymers. In the fast shear flow, entangled linear wormlike micelles appear to align, stretch, and tumble just like polymeric chains.

We report on recent progress in our understanding of the rheological behavior of polymer melts, resulting from the work of several investigators, mostly on fast extensional flows. They contributed new data from cleverly conceived experiments, new results of Brownian and molecular dynamics simulations, and suitable theoretical models. The outcome of such collective effort is the plausible certainty that flow-induced coalignment of the Kuhn segments of the polymer modifies the friction coefficient, which may depart considerably from the equilibrium value. It is also found that the effect is sensitive to the chemistry, being different in different polymers. It so appears that the universality of the rheological behavior of polymer melts, well documented under equilibrium conditions and in relatively slow flows, is lost in fast ones. This feature, however, opens interesting perspectives of future research and application.

We here extend to integral constitutive equations of the Doi–Edwards type the analysis of shear banding instabilities during shear startup recently performed by Moorcroft and Fielding [J. Rheol. 58, 103–147 (2014)]. Shear banding is the instability encountered in simple shear flows, whereby different shear rates coexist along the sample thickness, so that the velocity profile departs from the usually assumed linear profile. Results of both linear and nonlinear stability analysis mostly confirm those obtained by Moorcroft and Fielding [J. Rheol. 58, 103–147 (2014)] with the Rolie–Poly equation [Likhtman, A. E., and R. S. Graham, J. Non-Newtonian Fluid Mechan. 114, 1–12 (2003)] (which is an approximate differential counterpart of the Doi–Edwards integral one “[The Theory of Polymer Dynamics (Clarendon, Oxford, 1986)]”), but some significant differences are also found. Features that coincide with those predicted by the Rolie–Poly equation include: (i) For the nonmonotonic flow curve of the original Doi–Edwar...

We investigate the nonlinear shear and uniaxial extensional rheology of entangled polystyrene (PS) melts and solutions having the same number Z of entanglements, hence identical linear viscoelasticity. While experiments in extensional flows confirm that PS melts and solutions with the same Z behave differently, respective transient and steady data in simple shear over the largest possible range of rheometric shear rates (corresponding to Rouse−Weissenberg numbers from 0.01 to 40) demonstrate that melts and solutions exhibit identical behavior. Whereas the differences between melts and solutions in elongational flows are due to alignment-induced friction reduction (more effective in melts than in solutions), in shear flows they disappear since the rotational component reduces monomeric alignment substantially. Recent molecular dynamics simulations of entangled polymers show that rotation induces molecular tumbling at high shear rates, and here a tube-based model involving tumbling effects is proposed in or...

Recent data of unentangled polystyrene melts in startup elongational flow and subsequent relaxation [Matsumiya, Y.;Watanabe, H.;Masubuchi, Y.;Huang, Q.;Hassager, O.Nonlinear Elongational Rheology of Unentangled Polystyrene and Poly(p-Tert-Butylstyrene) Melts.Macromolecules2018,51(23), 9710−9729] are here compared to the predictions of Brownian simulations of Fraenkel chains endowed with a monomeric friction coefficient that decreases with increasing the order parameter of the Kuhn segments. The model compares favorably with all startup data if a time delay in reducing the monomeric friction coefficient is accounted for in the fastest flows, a feature already observed by Matsumiya et al. The same model also describes the relaxation data except for a minor discrepancy in the longest relaxation time, probably due to the imperfect monodispersity of the sample and/or to a minor influence of topological interactions, not strong enough to be called entanglements. A discrepancy is also observed at the beginning of relaxation after fast flows, the origin of which remains unexplained.

Shear thickening of telechelic associating polymers, particularly of hydrophobically modified ethoxylated urethanes (HEURs), is classically attributed to either non-Gaussian behavior of the chains, also called finite extensible nonlinear elasticity (FENE), or to flow-enhanced formation of network strands; however, in a recent paper [Macromolecules, 45, 888 (2012)], Suzuki et al. show that neither of the previously mentioned interpretations can hold true, as they find shear thickening while at the same time the network structure has essentially remained the equilibrium one, except for the fact that reassociation to the network of the strands that continuously detach from it is anisotropically enhanced in the shear gradient direction. In the present paper, we propose a different mechanism that might explain shear thickening, namely, the stress contribution arising from the repulsive interaction between the flowerlike micelles that are forced to interpenetrate one another by the shear flow. In support of thi...

We propose a simple, robust method to measure both the first and second normal stress differences of polymers, hence obtaining the full set of viscometric material functions in nonlinear shear flow. The method is based on the use of a modular cone-partitioned plate (CPP) setup with two different diameters of the inner plate, mounted on a rotational strain-controlled rheometer. The use of CPP allows extending the measured range of shear rates without edge fracture problems. The main advantage of such a protocol is that it overcomes limitations of previous approaches based on CPP (moderate temperatures not exceeding 120 °C, multiple measurements of samples with different volume) and yields data over a wide temperature range by performing a two-step measurement on two different samples with the same volume. The method was tested with two entangled polystyrene solutions at elevated temperatures, and the results were favorably compared with both the limited literature data on the second normal stress difference and the predictions obtained with a recent tube-based model of entangled polymers accounting for shear flow-induced molecular tumbling. Limitations and possible improvements of the proposed simple experimental protocol are also discussed.

Very recently, a new convective constraint release (CCR) single-mode (toy) model [Ianniruberto and Marrucci, J. Rheol. 58, 89–102 (2014)] has been proposed to account for the nowadays well-ascertained fact that flow induces some degree of disentanglement, the more so the faster is the flow [Baig et al., Macromolecules 43, 6886–6902 (2010)]. The previous work, successful in explaining some qualitative aspects of the nonlinear step strain response, is here extended to the multimode case by developing a model allowing for a spectrum of relaxation times in both orientational and stretch dynamics, the purpose being one of performing a quantitative comparison with literature data on nearly monodisperse linear polymers. Several data of relaxation after shear step strain and of time-dependent stress response in shear and elongational startup are considered. The overall agreement of the new multimode CCR model with data appears satisfactory.

Using the integral equation of the Doi–Edwards theory that only accounts for tube orientation of entangled linear polymers, we explore the behavior of the stress maximum typically observed in shear startup as a function of the waiting time tw between repeated startup runs. Depending on whether the first run is interrupted before the maximum, or sufficiently beyond it, the magnitude of the peak in the second run comes out nonmonotonic with tw or else monotonically increasing, respectively. A similar behavior has been observed in several experiments and commonly attributed to structural changes of the entangled network. By emphasizing the role played by the mere orientation of the network strands in faithfully reproducing all the observed behavior, at least qualitatively, our results help to put things in better perspective.

The uniaxial extensional-flow data of Huang and co-workers [ACS Macro Lett. 2013, 2, 741; J. Rheol. 2016, 60, 465] on several polystyrene systems, both melts and solutions, are here successfully compared with the predictions of a recent model [Ianniruberto, Macromolecules 2015, 48, 6306] that accounts for flow-induced friction-coefficient reduction. The model is here effectively simplified without loss of accuracy. Comparison with the solution data of Huang et al. [ACS Macro Lett. 2013, 2, 741], only differing for the size of the oligomeric solvent (from nearly one Kuhn length up to a few ones), reveals that friction reduction must include oligomer-size-dependent orientational coupling between the oligomeric solvent and the polymer molecules, such interactions vanishing for the smallest oligomer. The rheological “measure” of this nematic-like interaction as a function of the oligomer size (for small oligomers) is the novel result of the present paper, together with a further confirmation of the importance...

The classical Doi–Edwards theory for the steady-state stress of entangled linear polymers in uniaxial elongational flow is here extended to entangled branched polymers, specifically to stars and po...

For entangled linear monodisperse polymers, uniaxial elongational flow behavior was examined with the primitive chain network (PCN) simulation, which was originally formulated for a network of Gaus...

Although some experiments have addressed individual chain dynamics in the entangled state, the behavior of the quantity measured in the experiments, and its relation to chain conformation, deserves further study. In our previous work using primitive chain network simulations at equilibrium (Masubuchi et al, Nihon Reoroji Gakkaisi (J. Soc. Rheol. Jpn.) 36, 181, 2008), we examined the relaxation of the chain extension, x, which is the maximum distance between segments of any given chain. In this supplementary study we performed simulations under step shear deformations, away from the equilibrium state. It was found that the x relaxation under deformation is qualitatively similar to that of the relaxation modulus, where slow relaxation modes are insensitive to deformation while the intensity of fast modes is enhanced. As a result the longest relaxation time of x was found to be the same as the equilibrium state, and different from that of stress. For what concerns the behavior of the fast modes, less damping was observed for the relaxation of x than that for stress.

The primitive chain network model of entangled polymers, where primitive chains and entanglements are assigned in real 3-D space, is a possible step in the multi-scale path from atomistic simulations to macroscopic constitutive equations. In any coarse-grained model of a polymeric system, however, determination of the basic parameters is a crucial step. By fitting the linear viscoelastic response of several melts of monodisperse linear polymers taken from the literature, we here determine the corresponding basic parameters of the primitive chain network model. We concentrate in particular on the value of the mean molar mass between entanglements, and on how such a mass is related to plateau modulus from our simulations. We consistently find values that fall around 60% of the entanglement molecular weight as obtained from Ferry's definition, and we envisage that such a result is due to Brownian fluctuations of the entanglement nodes.

Orientation and stretch of entangled polymers under large step shear deformations were investigated through primitive chain network simulations. In the simulations, entangled polymer dynamics is described by 3D motion of entanglements, 1D sliding of monomers along the chain, and creation/destruction of entanglements described by hooking/unhooking with surrounding chains at chain ends. In addition to the conventionally proposed relaxation mechanisms that are reptation, contour length fluctuations, and constraint release (both thermal and convective), the simulations also account for force balance among entanglement strands converging to an entanglement node, and nodes also fluctuate in space. Nonlinear step strain data for monodisperse polystyrene melts (Ferri and Greco, Macromolecules, 37:5931, 2006) were quantitatively reproduced by using the same two molecular-weight-independent parameters already adopted by Masubuchi et al. (J Non-Newt Fluid Mech, 149:87, 2008) to fit linear viscoelastic data of several monodisperse polystyrene melts. Analysis of the orientation tensor and of the chain stretch ratio indicates that the segment orientation and stretch realized in the simulation are quantitatively described by a simple three-chain model (Marrucci et al., Macromol Symp, 158:57, 2000a).

Conformational relaxation of individual chain molecules in the entangled state has already been investigated by direct observation in fluorescent DNA experiments. Yet, possible connections between the quantity accessible in those experiments and well defined conformational measures have not been established so far. In this study, the relaxation behavior of the individual polymer conformation in the entangled state is simulated through the primitive chain network model, and compared to well known relaxations like that of stress or electric polarization. Inspired by the experiments, we use as a measure of the individual conformation the maximum distance between segments of a given polymer chain, which is often referred to as chain extension, and indicated by the symbol x. The relaxation behavior is calculated as auto-correlation function at equilibrium, suitably ensemble averaged to eliminate the noise. It is found that such x relaxation is significantly different from both stress relaxation and end-to-end dielectric relaxation, though the power-law exponent of the longest relaxation time vs. chain length appears to be the same for both the stress and the x measure.

A coarse-grained molecular simulation for block copolymers in the entangled state is proposed as an extension of the primitive chain network model. Polymers are represented as a sequence of segments between consecutive entanglements, the latter being modeled as sliplinks with other chains. Each sliplink connects two chains only, i.e., entanglements are taken as ‘binary’. The resulting 3D structure is a network of primitive chains, which makes our model different from other sliplink single-chain models, where the link to other chains is ‘virtual’. The 3D nature of our simulation makes it similar to, though considerably more coarse grained than, conventional molecular dynamics simulations. Because of the 3D space assignment of the polymers, monomeric density fields can be defined, and interactions due to different chemistry of the monomers can be accounted for, similarly to density field calculations. Polymer motion is described both by the 3D motion of sliplinks, and by the 1D transport of monomers along the primitive chain, while network topological rearrangement occurs due to chain-end hooking and unhooking processes. Each kinetic equation accounts for elastic forces along the chains, field forces arising from density gradients, and thermal random forces. In this paper, the primitive chain network model was modified (i) in the procedure of network rearrangement to account for the different chemistries in the copolymer, and (ii) in some details of the kinetic equations whenever the boundary between blocks is involved. We report results for diblock copolymers where for simplicity all relevant properties of the two monomers are the same, except for the interactions. Simulations reasonably reproduce the micro-phase formation process and the phase diagram for well entangled copolymers with a low calculation cost.

Motivated by recent data of Hassager and coworkers [A. Bach, K. Almdal, H.K. Rasmussen, O. Hassager, Elongational viscosity of narrow molar mass distribution polystyrene, Macromolecules, 36 (2003) 5174–5179], we develop a new tube model describing the non-linear behaviour of entangled monodisperse linear polymers. Within the context of well established tube theories, the model accounts for the effect of flow on tube diameter, thus somehow picking up a long standing suggestion by Wagner and Schaeffer [M.H. Wagner, J. Schaeffer, Non-linear strain measures for general biaxial extension of polymer melts, J. Rheol., 32 (1992) 1–26] among others. Since tubes with a deformation dependent cross section are rather difficult to deal with, we here limit model development to an artificial two-dimensional situation. The simple 2D model has the advantage of illustrating the new physical assumptions more transparently, and it already proves to correctly predict most qualitative features typically shown by shear and elongational data in the non-linear range.

Brownian dynamics simulations of the linear viscoelastic response of entangled polymers have been performed, and compared quantitatively with some existing solution data at a fixed concentration and variable molecular weigth. The model is a three-dimensional network where the nodes are sliplinks connecting chains in pair. The simulations make use of Langevin equations both for the node motion in space, and for the one-dimensional monomer sliding through sliplinks. Comparison with data is very satisfactory, but the molecular weigth between entanglements that emerges from the model is unconventionally small.

A single-mode constitutive equation is proposed which accounts, in a very simple way, for the most important feature of the molecular theory for entangled polymers based on the 'tube' concept, i.e. that the orientation of the tubes and the stretching of the chain within them relax over different time-scales. The much faster relaxation of stretch is essentially a Rouse process, whereas orientational relaxation takes place by a combination of several factors, including reptation, tube-length fluctuations and constraint release, both thermal and convective. The model presented here accounts for all of these mechanisms, albeit in a simplified way, with the exception of fluctuations, which cannot be dealt with in a single-mode theory. Extension to a multi-mode version should, however, be easy.

We study the rheometrical and complex flow response of the double-convection-reptation (DCR) model with chain stretch proposed recently by Ianniruberto and Marrucci (2002) for entangled linear polymers. The single- and two-mode differential versions of the model are used, with the parameter values identified by Ianniruberto and Marrucci (2002) for a nearly monodisperse polybutadiene solution. These authors found that the DCR model with stretch predicts the rheometrical shear behavior of the fluid well in the modest experimental range of deformation rates. Our calculations for the higher shear rates reached in simulations of complex flow reveal anomalous or questionable behavior, namely, shear thickening over an intermediate range of shear rates and large chain stretch in fast shear flows. This behavior is shown to be shared by the original integro-differential DCR theory, of which the differential DCR model is actually a mathematical approximation. We also show that the original DCR theory with stretch predicts excessive shear thinning at high shear rates, while its differential approximation remains stable for all shear rates. Using the backward-tracking Lagrangian particle method [Wapperom et al. (2000)], we investigate the response of the differential DCR model in start-up flow through an axisymmetric contraction/expansion geometry. We compare the single- and two-mode model predictions (in terms of the steady-state vortex structure, chain stretch, and overall pressure drop), and correlate these with the steady and start-up rheometrical responses in shear and extension. Significant chain stretch is predicted in the vicinity of the axis of symmetry and in thin boundary layers located at the constriction wall. As a result, the DCR predictions significantly depart from the stress-optical rule in these flow regions. Chain stretch also affects the flow kinematics, with the appearance of a large upstream steady-state vortex. Surprisingly, however, the predicted pressure drop is not affected much by these kinematical changes, and is, qualitatively described by a simple inelastic, shear-thinning model. (C) 2003 The Society of Rheology.

Parallel and orthogonal superposition experiments by Vermant et al. (1998) on a polydisperse, entangled polymeric solution are here analyzed by using a simple, multi-mode differential constitutive equation based on the tube model, and also accounting for convective constraint release effects. Model predictions are in very good qualitative and quantitative agreement with parallel superposition data, while some discrepancies are found with orthogonal data, thus suggesting that orthogonal superposition experiments represent a more severe test for molecularly-based constitutive equations.

The chapter contains a description of the observed behavior of polymeric liquids undergoing both slow and fast flows, mostly in the two basic situations of shear and uniaxial elongation. Next, the simplest phenomenological equations that somehow describe the complex flow behavior of polymeric liquids are presented. Finally, I show how similar equations have been derived based on suitable models of the polymeric molecules, and of their interactions. Connections to problems encountered in processing are made throughout the chapter.

Significant progress made in recent times on the basic molecular theory of Doi and Edwards for entangled polymers is briefly reviewed. In particular we present a recent version of the theory that is entirely described by a set of differential equations, and is therefore especially useful for simulations of complex flows as encountered in polymer processing. The model also provides an expression for the excess free energy that can be of use to predict the onset of flow-induced crystallisation.

We present a new multi-chain Brownian dynamics simulation of a polymeric network containing both crosslinks and slip-links (entanglements). We coarse-grain at the level of chain segments connecting consecutive nodes (cross- or slip-links). Affine displacement of nodes is not imposed; rather, their displacement as well as sliding of monomers through slip-links is governed by force balances. The stress-strain response in uniaxial extension is compared with the slip-link theories of Ball et al. (Polymer, 22 (1981) 1010) and Edwards and Vilgis (Polymer, 27 (1986) 483), and with the molecular-dynamics simulations of Grest et al. (J. Non-Cryst. Solids, 274 (2000) 139). Qualitative agreement both with the Mooney-Rivlin expression and with the stress upturn at large strains confirms the role of entanglements in explaining departure from the classical theory of phantom chain networks. However, quantitative agreement with data is satisfactory at low strains only, and the observed discrepancy at larger strains suggests possible refinements of the model. Additivity of free energy contributions of crosslinks and entanglements used in several molecular theories of rubber elasticity is confirmed by the simulation results.

The model recently proposed by the authors [J. Rheol., 45 (2001) 1305–1318] to account for convective constraint release (CCR) and chain stretch in entangled polymers is here extended to a multiple mode situation. Such an extension, indispensable for the polydisperse case, is believed to be important also for monodisperse polymers, especially in view of chain-end fluctuations. The model is compared with existing steady and transient start-up data of shear flows. It is also shown how the model can help explaining the frequently observed power-laws, especially for the case of melts.

In the classical theory of Doi and Edwards for entangled polymers the strain measure Q arises from the assumption of affine deformation of the tube segments. On the other hand, we have recently argued [G. Marrucci, F. Greco, and G. Ianniruberto, “Possible role of force balance on entanglements,” Macromol. Symp. (in press)] that force balance on the nodes of the entangled network generates departures from affinity. An ad hoc “lattice” model of the network was then proposed to fulfill force balance automatically, at least for single-step deformations. The corresponding strain measure is linked to the square root of the Finger tensor, and is therefore easy to use. Predictions obtained with the new strain measure are here compared with a variety of step strain data.

The molecular origin of the stress in entangled polymers is reconsidered here, in order to ascertain whether or not the pressure that the confined chain exerts on its 'tube of constraints' should be accounted for. Indeed, the classical expression for the stress includes only traction along the chain. The analysis shows that the pressure terms significantly contribute to the stress provided flow or deformation creates a long-lasting local anisotropy of the chain confinement. The effects of such anisotropy is described by means of elliptical cross-sections of the constraining tubes. The new expression for the stress tensor thus obtained properly degenerates into the classical one for circular cross sections. It is also shown that the stress-optical law remains obeyed. (C) 1998 Elsevier Science B.V. All rights reserved

Two-dimensional simulations of the Brownian dynamics of polymers in a grid of topological obstacles were carried out in this study. A first model made use of the exact expression for the free energy of a chain with fixed ends interacting with an obstacle and can be used to derive the elastic force acting on the subchains of a Rouse-like model. With this model we calculated the diffusion coefficient of chains of different lengths constrained by the fixed obstacles. A second model, less rigorous but more efficient computationally, was used to simulate the chain dynamics in fast flows. In such a case, the obstacles were convected according to the shear gradient. During the simulation, the configurational changes of the chain constrained by the obstacles could be monitored and the stresses which developed were calculated. These simulations reveal important effects of the conformational fluctuations at high shear rates.

Recent experiments by Li and Wang (Macromolecules 43:5904–5908, 2010) on parallel superposition flows of nearly monodisperse entangled polymer solutions revealing a flow-induced acceleration of the relaxation dynamics are here analyzed by using a simple differential constitutive equation based on the tube model to examine the role of convective constraint release (CCR) in these situations. Contrary to the expectations of Wang and coworkers, we find that such a flow-induced acceleration is not a CCR effect. Rather, our results suggest that the acceleration is simply due to convection per se (not to be confused with CCR) and to nonlinear orientation effects.

New data of shear startup on branched styrene-butadiene random (SBR) copolymers are reported, where the novelty consists in repeating the startup run after different rest times at zero stress. Here, the aim is one of exploring the “damage” introduced by the first run, as well as the subsequent recovery, if any, upon waiting increasingly long times. Differently from a linear sample, our branched melts show multiple peaks during the first run, as previously reported by Bacchelli [Kautschuk Gummi Kunststoffe 61, 188–191 (2008)] for similar SBR samples, and, more recently, by Snijkers et al. [ACS Macro Lett. 2, 601–604 (2013)] for a well-characterized comblike polystyrene melt. The repeated runs show an intriguing novel feature with respect to the case of linear polymers, namely, the first peak goes up initially, instead of down. The second peak goes down and seemingly recovers only after an extremely long time, longer than the largest relaxation time practically accessible to linear viscoelasticity, the latter not reaching the terminal behavior. All such features of nonlinear viscoelasticity of highly branched polymers are interpreted by using a simple theory inspired by the well-known pompom model.

It is widely accepted that the nonlinear viscoelasticity of polymers with long chain branching can be described by the pom-pom theory [J. Rheol. 1998, 42, 81] that accounts for branchpoint withdrawal (BPW) as a nonlinear relaxation mechanism of the backbone. In spite of the remarkable success attained by refined theories derived from the original pom-pom model, there remain a few questions on the consistency with the theoretical development for linear polymers. For instance, convective constraint release (CCR) is neglected in the pom-pom theories. In this study, primitive chain network simulations were performed to investigate the details of molecular motion under uniaxial elongation. The simulation automatically includes thermal and convective constraint release via the multichain dynamics. The code that was assembled with finitely extensible nonlinear elasticity (FENE), BPW and the stretch-orientation/induced friction reduction (SORF) shows a reasonable agreement with literature data of linear viscoelas...

The molecular mechanism known as convective constraint release (CCR) is here revisited to account for the fact that in fast flows topological entanglements decrease in number, as recently shown by the molecular dynamics simulations of Baig et al. [Macromolecules 43, 6886–6902 (2010)] and even before by the Brownian simulations of Yaoita et al. [J. Chem. Phys. 128, 154901 (2008)]. Based on the same Brownian dynamics code, Furuichi et al. [J. Chem. Phys. 133, 174902 (2010)] suggested that the time-strain separability in step strain relaxation is influenced by the entanglement density dynamics, an idea previously explored by Archer et al. [Macromolecules 35, 10216–10224 (2002)]. The simple model that is here proposed is therefore tested against literature data of stress relaxation following large step strains, where the time-strain separability is approached in a time much longer than the expected Rouse time, a mystery that has long remained unsolved. The new CCR theory gives a simple mathematical form to the suggestion of Furuichi et al. [J. Chem. Phys. 133, 174902 (2010)] and Archer et al. [Macromolecules 43, 6886–6902 (2002)] and also, for the first time, predicts a minimum in the time-dependent damping function, in agreement with the experiments.

It is known that polystyrene melts behave anomalously in fast elongational flows insofar as the steady-state elongational viscosity keeps decreasing with increasing stretching rate e, without show...