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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...

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.

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.

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.

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.

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 general solution for squeeze flow between closely-spaced plates of arbitrary shape includes an in-plane potential flow whose components satisfy the Cauchy–Riemann conditions, and the velocity field and the pressure are both determined by the boundary conditions at the edge of the plates. In contrast, the velocity field for the infinite-plate limit only requires boundary conditions at the surfaces of the plates. The infinite-plate problem is singular, and makes sense only as a limit of a sequence of flows in finite geometries, each of which has a well-defined coordinate origin.

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.

The dynamics of stiff polymers in the liquid or liquid–crystalline state is usually described by the rigid rod model. To account, at least to a first approximation, for the partial flexibility of the actual chain, the slightly-bending-rod model has been recently proposed, and a simplified two-dimensional version of the model has been developed in some detail. In this paper, we consider the full three-dimensional dynamic equation which, in its general form, is too complex for predictive purposes. We then derive a reduced form of the dynamic equation on the basis of the fact that some of the configurational variables have a much faster dynamics than others. The mathematical structure of the reduced form, though still complex, is close enough to that of the rigid rod model to allow for predictive calculations.

It is argued that the flow directly contributes a mechanism by which the topological obstacles are renewed. As a consequence, the chain diffusivity is increased (or, equivalently, the friction coefficient is decreased) in fast flows. Such a nonlinear effect is incorporated in a simple dumbbell model so as to produce a constitutive equation in closed form. An extension to the case of multiple relaxation times is also briefly considered. The predictions compare favourably with the Cox-Merz rule as well as with the behaviour of polymer melts in elongational flows.

In this paper we derive a few analytical results for the case of polymeric solutions flowing past a spherical obstacle. The calculations are meant to ascertain the possible importance of flow-induced migration (i.e., of changes in concentration), making use of the recent two-fluid theory of Doi and Milner. In order to obtain solutions in closed form, the problem is suitably linearised; in particular, we have considered both the case of Newtonian behaviour of the polymeric liquid flowing past the sphere, and that of a linearly elastic polymeric gel perturbed by a localised force. The results, though confirming migration effects, indicate that, in these linear limits at least, the concentration change decays quadratically with the reciprocal distance from the centre of the perturbation, i.e., no long “wake” is predicted. Another outcome of these calculations refers specifically to the viscous case. The classical Stokes result is recovered only as a singular solution of the two-fluid problem.

The primitive chain network (PCN) model successfully employed to simulate the rheology of entangled polymers is here tested versus less coarse-grained (lattice or atomistic) models for what concerns the structure of the network at equilibrium (i.e., in the absence of flow). By network structure, we mean the distributions of some relevant quantities such as subchain length in space or in monomer number. Indeed, lattice and atomistic simulations are obviously more accurate, but are also more difficult to use in nonequilibrium flow situations, especially for long entangled polymers. Conversely, the coarse-grained PCN model that deals more easily with rheology lacks, strictly speaking, a rigorous foundation. It is therefore important to verify whether or not the equilibrium structure of the network predicted by the PCN model is consistent with the results recently obtained by using lattice and atomistic simulations. In this work, we focus on single chain properties of the entangled network. Considering the significant differences in modeling the polymer molecules, the results here obtained appear encouraging, thus providing a more solid foundation to Brownian simulations based on the PCN model. Comparison with the existing theories also proves favorable.

On the basis of the rod-like polymer theory of the nematic phase, originally formulated by Doi, an attempt is made to develop a constitutive equation for the stress tensor which does not require a preliminary determination of the distribution function of rod orientations and yet is capable of predicting well-known peculiarities of the rheological behaviour of many liquid crystalline polymers, such as the negative normal stress effect. In view of the intrinsic difficulties of this task, the work developed in this paper is fictitiously restricted to a shear flow in two dimensions, with the aim of understanding, in this relatively simple case, what kind of approximations are permissible. It is found that closure approximations at the lowest possible level are inadequate, whereas a suitable closure at the next level works satisfactorily.

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

The two‐dimensional model previously considered is here used to obtain predictions in the complex situation of the tumbling regime which prevails at low shear rates. Although each domain is permanently in a time‐dependent, periodic regime, the macroscopic response can be stationary in time because of the well‐known polydomain structure. First, the average steady rheological response for this situation is calculated by taking a polydomain structure which neglects interdomain interactions. Although the steady state predictions thus obtained favorably compare with the experimental results throughout the range of shear rates, the transient start‐up responses do not, because of the crucial role played by the interactions in such a case. These interactions, due to Frank elasticity, are then introduced in the model in the simplest possible way, i.e., by use of a mean field potential. Under this assumption, also the predictions of transient behavior in the tumbling regime show the correct qualitative features.

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.

We propose a simple way of including chain stretch effects in convective constraint release theories for entangled polymers. The main idea is that the characteristic time of orientational relaxation depends in a series-parallel way on all three relevant mechanisms, i.e., reptation, constraint release (thermal and convective), and Rouse relaxation. As usual, a separate equation describes chain stretch, which however is assumed not to be affected by constraint release. The model is further simplified by writing the orientational equation in differential form. For step strains, the successful damping function of the Doi–Edwards theory is exactly preserved. Predictions in steady shear also favorably compare with typical data of nearly monodisperse polymers.

The molecular model used by Doi and Edwards in developing their rheological theory admits a simple form for the free energy of deformation, at least when the independent alignment approximation is used. A general expression is derived, as well as the specific form which applies to an instantaneous shear deformation. The latter is used to analyze the possibility that instabilities, or “deformational phase separations,” arise in the course of stress relaxation experiments. The results of the analysis are compared with recent experimental results by Osaki and Kurata and by Vrentas and Graessley which show an anomalous relaxation behavior.

Predictive equations for terminal velocities of liquid drops in Newtonian and non-Newtonian systems are reviewed. The wave theory analogy is extended to include liquid-liquid systems. Experimental data obtained with seven different systems are presented. The agreement with predictive equations is satisfactory. Interface rigidity was observed in all the systems investigated.

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.

A possible description of the viscoelastic behaviour of polymer melts in the non-linear r’ange which has been recently considered makes use of stress equations of the form $$ \mathop{\sigma }\limits_{ \sim } (t) = \int_{{ - \infty }}^{t} {\dot{\mu }(t - t')\mathop{Q}\limits_{{ \sim t}} } (t')dt' $$ (1) where the tensor \( \mathop{Q}\limits_{{\sim t}} (t') \) is determined by the deformation occurred between t’ and t and the memory kernel \( \dot{\mu } \) is a function of time only.