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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.

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.

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.

The classical dumbbell and Rouse models for dilute polymer solutions are here modified by introducing an anisotropic diffusivity which mimics the effect of topological interactions arising in polymeric solutions in the vicinity of the overlap concentration c ∗ . Closed-form constitutive equations are derived by making use of typical pre-averaging approximations. Predictions in steady and transient shear flows for tangential and normal stresses are compared to data recently obtained by Osaki et al. (J. Polym. Sci.: Polym. Phys. 38 (2000) 1917), on a polystyrene solution close to c ∗ . As regards the second normal stress difference, more rarely measured, we compare the model predictions with data by Magda et al. (Macromolecules 26 (1993) 1696).

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

A recently suggested relaxation mechanism which is expected to be dominant in fast flows of polymer melts is combined here with the tube model of Doi and Edwards, and the corresponding constitutive equation is derived. The mechanism consists of the renewal of topology by convective displacement of entanglements. Detailed predictions for the flow curve in steady shear are obtained in order to test the agreement with the Cox-Merz rule. With respect to the basic tube model, significant progress is achieved.