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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 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 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...

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.

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...

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...

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...

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).

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.

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.

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.

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.

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 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

Molecular theories for polymer rheology are based on conformational dynamics of the polymeric chain. Hence, measurements directly related to molecular conformations appear more appealing than indirect ones obtained from rheology. In this study, primitive chain network simulations are compared to experimental data of entangled DNA solutions [Teixeira et al., Macromolecules 40, 2461 (2007)]. In addition to rheological comparisons of both linear and nonlinear viscoelasticities, a molecular extension measure obtained by Teixeira et al. through fluorescent microscopy is compared to simulations, in terms of both averages and distributions. The influence of flow on conformational distributions has never been simulated for the case of entangled polymers, and how DNA molecular individualism extends to the entangled regime is not known. The linear viscoelastic response and the viscosity growth curve in the nonlinear regime are found in good agreement with data for various DNA concentrations. Conversely, the molecular extension measure shows significant departures, even under equilibrium conditions. The reason for such discrepancies remains unknown.

The mechanical properties of a polymeric network containing both crosslinks and sliplinks (entanglements) are studied using a multi-chain Brownian dynamics simulation. We coarse-grain at the level of chain segments connecting consecutive nodes (cross- or sliplinks), with particular attention to the Gaussian statistics of the network. Affine displacement of nodes is not imposed: their displacement as well as sliding of monomers through sliplinks is governed by force balances. The simulation results of stress in uniaxial extension and the full stress tensor in simple shear including the (non-zero) second normal stress difference are presented for monodisperse chains with up to 18 entanglements between two crosslinks. The cases of two different force laws of the subchains (Gaussian chains and chains with finite extensibility) for two different numbers of monomers in a subchain (no = 50 and no = 100) are examined. It is shown that the additivity assumption of slip- and crosslink contribution holds for sufficiently long chains with two or more entanglements, and that it can be used to construct the strain response of a network of infinitely long chains. An important consequence is that the contribution of sliplinks to the small-strain shear modulus is about ⅔ of the contribution of a crosslink.

A novel simulation method that can calculate the long-time response of polymeric liquids in the entangled state is described. The polymer chain is replaced by a sequence of subchains connecting consecutive entanglements, called the 'primitive' chain. Collectively, the primitive chains form a rubberlike network, the nodes of which are the entanglements. The dynamics is modelled separately in two parts: motion of the entanglements in space and motion of monomers along the primitive chain (reptation). Hence, in contrast to other long-time simulations based on the tube model, and similarly to conventional short-time molecular dynamics, the real space arrangement of the chains is accounted for. In this paper, extensions of the model towards fast flows, large deformations, branched polymers and polymer blends are introduced, in order to obtain simulations of polymeric systems with a variety of molecular structures over the longest relaxation time of the system. Quantitative tests against existing experimental data indicate good agreement with just two adjustable parameters, which are the scale factors in space and time.

The concept of dynamic tube dilation (DTD) is here used to formulate a new simulation scheme to obtain the linear viscoelastic response of long chains with a large number of entanglements. The new scheme is based on the primitive chain network model previously proposed by some of the authors, and successfully employed to simulate linear and nonlinear behavior of moderately entangled polymers. Scaling laws are generated by the DTD concept, and allow for prediction of the linear response of very long chains on the basis of suitable simulations performed on shorter ones, without introducing adjustable parameters. Tests of the method against existing data for linear monodisperse polyisoprene and polystyrene show good quantitative agreement.

Recent data by Hassager and co-workers [Bach et al. Macromolecules 2003, 36, 5174] of elongational viscosity of nearly monodisperse polystyrene melts are interpreted by including in the classical tube theories for entangled polymer dynamics an interchain repulsive contribution. The proposed theory predicts the observed power law of ηel vs e in a straightforward way and qualitatively explains the observed scaling with the polymer molecular mass. Possible generalizations are discussed.

The viscosity of polymeric liquids crucially depends on polymer concentration c and molar mass M. These dependencies can generally be predicted from relatively simple theories, because what really matters is the chain-like structure of the polymer molecule rather than its detailed chemistry. In this chapter we summarize the main concepts leading to these predictions throughout the concentration range, i.e., from dilute solutions up to polymer melts. However, only the so-called zero-shear viscosity will be considered because the nonlinear effects arising from the coupling between flow and molecular “structure” are more complex and fall outside the scope of this chapter. In dilute solutions the important concept is that of “intrinsic” viscosity which, through dependence on M, reveals structural features of the polymer (flexible chain vs. rigid rodlike, for example) as well as the solvent quality. In semidilute solutions of long polymers, the overall structure of the system becomes that of an impermanent network of entangled chains. The viscosity is then related to the elasticity of the network and to the kinetics of chain disengagement. Typically, the viscosity scaling takes the form of power laws in both M and c. The M dependencies remain the same in the particularly relevant case of polymer melts. The chapter ends with a brief description of systems with localized interactions (or sticky points) in which the viscosity is particularly sensitive to the strength of such interactions.

The basic concepts of entangled polymer dynamics, like that of ‘tube’ of constraints and of ‘reptation’, have not yet exhausted their momentum. Further progress has taken place recently with detailed calculations of tube length fluctuations and constraint release, not only for branched polymers but also for linear ones. Progress in the non-linear range has occurred with the concept of convective constraint release and, for branched polymers, with the so-called ‘pom-pom’ model. Lack of complete agreement between theory and experiments stimulates work to proceed along various lines.

The flow-induced concentration changes in polymers are analysed using the two-fluid theory of Doi and Milner, which predicts migration in flows with either curvilinear trajectories or non-uniform shear rates. Here we show that, even in simple shear flows, migration can be relevant since it can make the stationary solution with uniform concentration linearly unstable in fast flows.

To understand the flow behaviour of polymeric liquids, it is important to analyse the microscopic origin of the stress. The classical molecular theories relate the stress to the tensile force of entropic origin acting along the chains. The momentum transfer due to collisions between different molecules is not explicitly considered. Though certainly adequate for dilute solutions, that assumption becomes questionable in concentrated solutions and melts

The dynamics of concentrated polymers of high molecular mass is dominated by topological interactions (or entanglements). The nonlinear effects shown by these systems in fast flows are still poorly understood, however. We have recently noted that a mechanism of convective constraint release should play a significant role in the nonlinear range, and preliminary calculations have been made both for a simple dumbbell model and for the tube model of Doi and Edwards Here we show that further elaboration of the simple dumbbell model can predict quite complex rheological behaviours such as those observed by Navard and coworkers in semi-dilute solutions of ultrahigh molecular mass polymers.

Slit devolatilization of polymers has been considered in a recent work by Maffettone et al. (1991), who have presented a successful model for single slit devolatilization. In this article, the stability of operation of a realistic multislit process is considered. It is shown that the flow rate per slit which can be stably operated may be significantly less than the optimal one for single slit operation. Stability improves as the concentration of volatiles in the feed solution decreases. The non-Newtonian character of the feed solution at high shear rates is discussed in qualitative terms.

The migration phenomenon in polymeric solutions undergoing flow in straight channels is analyzed by using the recent two-fluid theory of Doi and Milner, so far applied to shear flows in rotating devices. Migration is predicted to take place along the transversal direction of the Poiseuille flow (from the walls of the channel toward the center) if a constitutive equation appropriate to the entangled state of polymers is used. More specifically, we considered the behavior of semidilute solutions in good solvents. The detailed formulation of the boundary value problem of the slit flow reveals an unusual boundary condition at the wall. The governing equations are solved in closed form for the steady state and numerically for the transient. It is shown that, although a stationary concentration profile is achieved only after a very long time, effects of the migration phenomenon close to the wall can become significant much sooner. Outlooks for future work are discussed.