Primitive Chain Network Simulations Research Articles

Primitive Chain Network Simulations for Double Peaks in Shear Stress under Fast Flows of Bidisperse Entangled Polymers.

A few experiments have reported that the time development of shear stress under fast-startup shear deformations exhibits double peaks before reaching a steady state for bimodal blends of entangled linear polymers under specific conditions. To understand this phenomenon, multi-chain slip-link simulations, based on the primitive chain network model, were conducted on the literature data of a bimodal polystyrene solution. Owing to reasonable agreement between their data and our simulation results, the stress was decomposed into contributions from long- and short-chain components and decoupled into segment number, stretch, and orientation. The analysis revealed that the first and second peaks correspond to the short-chain orientation and the long-chain stretch, respectively. The results also implied that the peak positions are not affected by the mixing of short and long chains, although the intensity of the second peak depends on mixing conditions in a complicated manner.

Anisotropy of Shear Relaxation in Primitive Chain Network Simulations for Entangled Polymers Confined between Flat Walls

Anisotropy of Shear Relaxation in Primitive Chain Network Simulations for Entangled Polymers Confined between Flat Walls

Primitive chain network simulations of the nonlinear rheology of polystyrene melts: Friction reduction and fluctuation-dissipation theorem

Primitive chain network simulations of the nonlinear rheology of polystyrene melts: Friction reduction and fluctuation-dissipation theorem

Tube Survival Fraction in Primitive Chain Network Simulations

In the framework of the tube model, the survival fraction of dilated tube, φʹ(t), is essential since the relaxation of the entire system is described according to this quantity. However, a method to define it in multi-chain simulations has yet to be considered. In this study, primitive chain network simulations were performed, and a few structural quantities were discussed as possible candidates for use as φʹ(t). Before analysis, the simulation was evaluated against literature data for linear polyisoprene melts, and quantitative agreement was obtained for viscoelastic and dielectric relaxations. Afterward, the end-to-end relaxation function was compared to survival probability of the slip link φs(t) and the ratio of monomers between the most distant surviving slip link pair to the total number of monomers ns(t)/nt. The result implies that ns(t)/nt is comparable to φʹ(t). Comparison to atomistic molecular dynamics (MD) simulations for polyethylene showed similar results, implying that φʹ(t) can be obtained even in MD simulations if the surviving entanglement points can be specified.

Elongational viscosity of poly(propylene carbonate) melts: tube-based modelling and primitive chain network simulations

In fast elongational flows, linear polymer melts exhibit a monotonic decrease of the viscosity with increasing strain rate, even beyond the contraction rate of the polymer defined by the Rouse time. We consider two possible explanations of this phenomenon: (a) the reduction of monomeric friction and (b) the reduction of the tube diameter with increasing deformation leading to an Enhanced Relaxation of Stretch (ERS) on smaller length scales. (Masubuchi et al. (2022) reported Primitive Chain Network (PCN) simulations using an empirical friction reduction model depending on segmental orientation and could reproduce the elongational viscosity data of three poly(propylene carbonate) melts and a polystyrene melt. Here, we show that the mesoscopic tube-based ESR model (Wagner and Narimissa 2021) provides quantitative agreement with the same data set based exclusively on the linear-viscoelastic characterization and the Rouse time. From the ERS model, a parameter-free universal relation of monomeric friction reduction as a function of segmental stretch can be derived. PCN simulations using this friction reduction relation are shown to reproduce quantitatively the experimental data even without any fitting parameter. The comparison with results of the earlier PCN simulation results with friction depending on segmental orientation demonstrates that the two friction relations examined work equally well which implies that the physical mechanisms of friction reduction are still open for discussion.

Analysis of Elongational Viscosity of Entangled Poly (Propylene Carbonate) Melts by Primitive Chain Network Simulations.

It has been established that the elongational rheology of polymers depends on their chemistry. However, the analysis of experimental data has been reported for only a few polymers. In this study, we analyzed the elongational viscosity of poly (propylene carbonate) (PPC) melts in terms of monomeric friction via primitive chain network simulations. By incorporating a small polydispersity of materials, the linear viscoelastic response was semi-quantitatively reproduced. Owing to this agreement, we determined units of time and modulus to carry out elongational simulations. The simulation with constant monomeric friction overestimated elongational viscosity, whereas it nicely captured the experimental data if friction decreased with increasing segment orientation. To see the effect of chemistry, we also conducted the simulation for a polystyrene (PS) melt, which has a similar entanglement number per chain and a polydispersity index. The results imply that PPC and PS behave similarly in terms of the reduction of friction under fast deformations.

A new perspective on monomeric friction reduction in fast elongational flows of polystyrene melts and solutions

Primitive chain network (PCN) and non-equilibrium molecular dynamics (NEMD) simulations indicate that the nonuniversality seen in the elongational rheology of linear polymer melts and concentrated solutions is caused by monomeric friction reduction. We present, in the context of tube models with varying tube diameter, a new constitutive equation based on enhanced relaxation of stretch (ERS) and find good agreement with experimental evidence and with predictions of the extended interchain pressure (EIP) model. The stretch evolution equations of the ERS and the EIP model can be converted in a form, which allows expressing them in terms of monomeric friction reduction. Monomeric friction coefficients obtained from the ERS model and those used in PCN and NEMD simulations show at least qualitative agreement. While the reduction of monomeric friction is incorporated in PCN and NEMD simulations by empirical correlations between friction coefficient and segmental orientation or elongational stress fitted to elongational viscosity data, the EIP and ERS models provide analytical and parameter-free relations of monomeric friction reduction as a function of chain stretch.

Primitive Chain Network Simulations of Entangled Melts of Symmetric and Asymmetric Star Polymers in Uniaxial Elongational Flows

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.

Elasticity of Randomly Cross-Linked Networks in Primitive Chain Network Simulations

Primitive chain network simulations for randomly cross-linked slip-link networks were performed. For the percolated networks, the stress-strain relationship was compared to the theories by Ball et al. [Polymer, 22, 1010 (1981)] and Rubinstein and Panyukov [Macromolecules, 35, 6670 (2002)]. The simulation results were reasonably reproduced by both theories, given that the contributions from cross-links and slip-links were used as fitting parameters. However, these parameters were model dependent. Besides, the theories cannot describe the simulation results if the parameters were determined from the number of active links involved in the percolated networks. These results reveal that the fitting of experimental data to the theories does not provide a fraction of entanglements in the system unless the network only consists of Gaussian strands and it correctly reaches the state of free-energy minimum.

Wall slip in primitive chain network simulations of shear startup of entangled polymers and its effect on the shear stress undershoot

In some recent experiments on entangled polymers of stress growth in the 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 the 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. Disentanglement of grafted chains from bulk ones, taking place throughout the stress overshoot region, does not contribute to the stress undershoot.

Effects of Constraint-Release on Entangled Polymer Dynamics in Primitive Chain Network Simulations

Effects of Constraint-Release on Entangled Polymer Dynamics in Primitive Chain Network Simulations

Primitive chain network simulations for the interrupted shear response of entangled polymeric liquids.

The non-linear viscoelastic response under interrupted shear flows is one of the interesting characteristics of entangled polymers. In particular, the stress overshoot in the resumed shear has been discussed concerning the recovery of the entanglement network in some studies. In this study, we performed multichain slip-link simulations to observe the molecular structure of an entangled polymer melt. After confirming the reasonable reproducibility of our simulation with the literature data, we analyzed the molecular characteristics following the decoupling approximation. We reasonably found that the segment orientation dominates the stress overshoot even under the resumed shear with minor contributions from the segment stretch and entanglement density. We defined the mitigation function for the recovery of the stress overshoot as a function of the rest time and compared it with the relaxation of the molecular quantities after the initial shear. As a result, we have found that the mitigation of the stress overshoot coincides with the relaxation of entanglement density.

Primitive chain network simulations for H-polymers under fast shear.

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.

Comparison among Multi-Chain Simulations for Entangled Polymers under Fast Shear

Due to the slow relaxation nature of polymers, attempts for the development of multi-scale simulations have been made towards the optimization of material processing concerning the molecular characteristics. However, transferability among the utilized models has not been frequently discussed. In this study, a few multi-chain simulations for entangled polymers were compared for the viscosity growth curve under fast shear flows. The multi-chain slip-spring (MCSS) simulations and the multi-chain slip-link simulations (also known as primitive chain network (PCN) simulations) were performed, and the results were compared to the data reported by Cao and Likhtman for the bead-spring model proposed by Kremer and Grest. The conversion parameters for the unit of time and stress were determined from the comparison for the linear viscoelasticity, for which the tree models are essentially transferrable. The obtained viscosity growth curve for the MCSS simulations was in good agreement with that reported for the bead-spring simulations, including the viscosity overshoot. For the PCN simulations, the magnitude of viscosity overshoot was underestimated, whereas the steady state viscosity was reasonably captured. The results demonstrate that the multi-scale strategy by the examined models would work in a certain extent.

Stress Undershoot of Entangled Polymers under Fast Startup Shear Flows in Primitive Chain Network Simulations

Stress Undershoot of Entangled Polymers under Fast Startup Shear Flows in Primitive Chain Network Simulations

Orientational cross correlations between entangled branch polymers in primitive chain network simulations.

Although it has not been frequently discussed, contributions of the orientational cross-correlation (OCC) between entangled polymers are not negligible in the relaxation modulus. In the present study, OCC contributions were investigated for 4- and 6-arm star-branched and H-branched polymers by means of multi-chain slip-link simulations. Owing to the molecular-level description of the simulation, the segment orientation was traced separately for each molecule as well as each subchain composing the molecules. Then, the OCC was calculated between different molecules and different subchains. The results revealed that the amount of OCC between different molecules is virtually identical to that of linear polymers regardless of the branching structure. The OCC between constituent subchains of the same molecule is significantly smaller than the OCC between different molecules, although its intensity and time-dependent behavior depend on the branching structure as well as the molecular weight. These results lend support to the single-chain models given that the OCC effects are embedded into the stress-optical coefficient, which is independent of the branching structure.

Primitive chain network simulations of probe rheology.

Probe rheology experiments, in which the dynamics of a small amount of probe chains dissolved in immobile matrix chains is discussed, have been performed for the development of molecular theories for entangled polymer dynamics. Although probe chain dynamics in probe rheology is considered hypothetically as single chain dynamics in fixed tube-shaped confinement, it has not been fully elucidated. For instance, the end-to-end relaxation of probe chains is slower than that for monodisperse melts, unlike the conventional molecular theories. In this study, the viscoelastic and dielectric relaxations of probe chains were calculated by primitive chain network simulations. The simulations semi-quantitatively reproduced the dielectric relaxation, which reflects the effect of constraint release on the end-to-end relaxation. Fair agreement was also obtained for the viscoelastic relaxation time. However, the viscoelastic relaxation intensity was underestimated, possibly due to some flaws in the model for the inter-chain cross-correlations between probe and matrix chains.

Orientational Cross-Correlation in Entangled Binary Blends in Primitive Chain Network Simulations

It has been reported from the molecular dynamics simulation that the orientational cross-correlation (OCC) has a significant contribution to the relaxation modulus of polymers. Remarkably, the OCC contribution has been found to be universal with respect to the molecular weight and even to its distribution for unentangled polymers [Cao; Likhtman Phys. Rev. Lett. 2010, 104, 207801]. In this study, the OCC in entangled bidisperse polymer melts was evaluated by multichain slip-link simulations. The obtained OCC was similar to that in monodisperse systems in its intensity. On the other hand, the time development reflected the motion of short and long chains, and consequently the universality was not observed when the molecular weights of two components are well-separated and the long chain fraction is small. Comparison to the results obtained by the other slip-link model suggests that the cross-correlation is induced by the force balance and the fluctuation at the entanglement.

Effect of Osmotic Force on Orientational Cross-correlation in Primitive Chain Network Simulation

Effect of Osmotic Force on Orientational Cross-correlation in Primitive Chain Network Simulation

Primitive chain network simulations for elongational viscosity of bidisperse polystyrene melts

In spite of the industrial significance, molecular mechanism of the strain hardening saliently observed in bidisperse polymeric liquids has not been elucidated yet. In this study, the multi-chain slip-link simulation (called primitive chain network simulation) was performed for the bidisperse polystyrene blends for which experimental data for elongational viscosity have been reported earlier. The simulation reasonably reproduced linear viscoelasticity and transient and steady uniaxial elongational viscosities. It has been confirmed that the long chain stretch dominates the stress at the strain hardening as already demonstrated earlier via the tube model. The molecular analysis employing the decoupling approximation revealed for the first time that there exist two molecular mechanisms to induce strain hardening in bidisperse blends. The mechanism switches depending on the Weissenberg number with respect to the Rouse relaxation time of the long chain, Wi RL. At Wi RL < 1, the simultaneous increase of the long chain orientation and stretch with increasing Wi RL lifts the viscosity beyond the Trouton’s viscosity. At Wi RL ≥ 1, the isotropic short chain suppresses the stretch/orientation-induced reduction of friction to enhance the stretch of long chain.