Bidisperse Blends Research Articles

Constraint Release Rouse Mechanisms in Bidisperse Linear Polymers: Investigation of the Release Time of a Short-Long Entanglement

Despite a wide set of experimental data and a large number of studies, the quantitative description of the relaxation mechanisms involved in the disorientation process of bidisperse blends is still under discussion. In particular, while it has been shown that the relaxation of self-unentangled long chains diluted in a short chain matrix is well approximated by a Constraint Release Rouse (CRR) mechanism, there is no consensus on the value of the average release time of their entanglements, τobs, which fixes the timescale of the CRR relaxation. Therefore, the first objective of the present work is to discuss the different approaches proposed to determine this time and compare them to a large set of experimental viscoelastic data, either newly measured (poly(methyl-)methacrylate and 1,4-polybutadiene blends) or coming from the literature (polystyrene and polyisoprene blends). Based on this large set of data, it is found that with respect to the molar mass of the short chain matrix, τobs follows a power law with an exponent close to 2.5, rather than 3 as previously proposed. While this slight change in the power law exponent does not strongly affect the values of the constraint release times, the results obtained suggest the universality of the CRR process. Finally, we propose a new description of τobs, which is implemented in a tube-based model. The accurate description of the experimental data obtained provides a good starting point to extend this approach to self-entangled binary blends.

Stretch-orientation-induced reduction of friction in well-entangled bidisperse blends: a dual slip-link simulation study

We investigated the rheological properties of bidisperse entangled-polymer blends under high-deformation-rate flows by slip-link simulations with a friction reduction mechanism. The friction reduction mechanism induced by the stretch and orientation (SORF) is important to predict the viscoelasticity under uniaxial elongational flows. To test the applicability of this mechanism for bidisperse systems, we incorporated an expression of friction reduction (Yaoita et al. Macromolecules 45:2773–2782 2012) into the Doi-Takimoto slip-link model (DT model) (Doi and Takimoto Philos Trans R Soc Lond A 361:641–652 2003). For six experimental bidisperse systems, i.e., four polystyrene blends and two polyisoprene blends, the extended DT model where the order parameter of the friction reduction mechanism is evaluated through the component averages succeeds in reproducing the data under uniaxial elongation and shear. This success is due to the suppression of the stretch of the longer chains using the statistical average over each component. Through this study, the SORF expression improves the rheological prediction for bidisperse entangled polymer melts under uniaxial elongational flows with strain rates comparable to or larger than the inverse of the Rouse relaxation time of the longer chains. Additionally, the predictions with the SORF using the component average for the stretches reproduce the steady viscosities because under elongational flows, the states of the components with different molecular weights clearly differ from each other depending on their Rouse relaxation time. The finding means that for chain dynamics, the friction coefficient is determined by the state of the surrounding polymer chains and the state of the chain.

Elongational viscosity and brittle fracture of bidisperse blends of a high and several low molar mass polystyrenes

Elongational viscosity data of four well-characterized blends consisting of 10% mass fraction of monodisperse polystyrene PS-820k (molar mass of 820 kg/mol) and 90% matrix polystyrenes with a molar mass of 8.8, 23, 34, and 73 kg/mol, respectively, as reported by Shahid et al. Macromolecules 52: 2521–2530, 2019 are analyzed by the extended interchain pressure (EIP) model including the effects of finite chain extensibility and filament rupture. Except for the linear-viscoelastic contribution of the matrix, the elongational viscosity of the blends is mainly determined by the high molar mass component PS-820k at elongation rates when no stretching of the lower molar mass matrix chains is expected. The stretching of the long chains is shown to be widely independent of the molar mass of the matrix reaching from non-entangled oligomeric styrene (8.8 kg/mol) to well-entangled polystyrene (73kg/mol). Quantitative agreement between data and model can be obtained when taking the interaction of the long chains of PS-820k with the shorter matrix chains of PS-23k, PS-34k, and PS-73k into account. The interaction of long and short chains leads to additional entanglements along the long chains of PS-820k, which slow down relaxation of the long chains, as clearly seen in the linear-viscoelastic behavior. According to the EIP model, an increased number of entanglements also lead to enhanced interchain pressure, which limits maximal stretch. The reduced maximal stretch of the long chains due to entanglements of long chains with shorter matrix chains is quantified by introducing an effective polymer fraction of the long chains, which increases with the increasing length of the matrix chains resulting in the excellent agreement of experimental data and model predictions.

Effect of particle angularity on flow regime transitions and segregation of bidisperse blends in a rotating drum

Effect of particle angularity on flow regime transitions and segregation of bidisperse blends in a rotating drum

Shear induced demixing in bidisperse and polydisperse polymer blends: Predictions from a multifluid model

In light of recent advancements in the constitutive modelling of bidisperse and polydisperse entangled linear polymers, we present a new multi fluid generalization of the classic two fluid approximation for flows of inhomogeneous polymer blends. As an application of the model, we consider predictions for the linear and nonlinear dynamics of shear induced demixing (SID) instabilities in blends with bidisperse and lognormal molecular weight distributions. We find that even in the absence of any chemical contrast between component chains, an imposed flow can induce a demixing instability provided there is sufficient contrast in the size of the two chains. The lower bound polydispersity for SID coincides with the point where elastic forces (kT per entanglement) scaled by the contrast between chains (e.g. polydispersity index minus one) exceed the entropic forces for mixing (kT per chain). For bi-disperse blends, we show that the non-linear dynamics of SID strongly resemble what has previously been shown for SID in entangled polymer solutions.

Modelling contraction flows of bi-disperse polymer blends using the Rolie-Poly and Rolie-Double-Poly equations

The flow of a bi-disperse polymer melt through a hyperbolic contraction is simulated using the recently proposed Rolie-Double-Poly constitutive model (Boudara et al., 2019). This simplified tube model takes account of the nonlinear coupling between the dynamics of the long and short-chains in a bi-disperse blend, in particular it reproduces the enhancement of the stretch relaxation time that arises from the coupling between constraint release and chain retraction. Flow calculations are performed by implementing both the Rolie-Double-Poly and multimode Rolie-Poly models in OpenFOAM using the RheolTool library. While both models predict very similar flow patterns, the enhanced stretch relaxation of the Rolie-Double-Pol models results in an increase in the molecular stretch of the long chain component in the pure extensional flow along the centre-line of the contraction, but a decrease in the stretch in shear-flow near the channel walls.

Nonlinear rheology of polydisperse blends of entangled linear polymers: Rolie-Double-Poly models

While there has been much success in modeling the linear and nonlinear rheology of monodisperse entangled linear polymers, progress in the constitutive modeling of polymeric materials continues to lag behind the needs of industry. Industrially sourced polymers are typically polydisperse (comprising a broad distribution of molecular weights), making their rheology more suitable for processing but also more difficult to predict. To date, there are no molecular-based constitutive models that are practically suitable for describing industrially relevant polymers in industrially relevant flows. In this article, we extend but strongly simplify the model of Read et al. [J. Rheol. 56, 823–873 (2012)], which is able to predict the linear and nonlinear rheology of bidisperse blends but is prohibitively complex for industrial use. We propose a pair of simplified tube models for polydisperse melts of entangled linear polymers that combine the success of the double reptation approximation [des Cloizeaux, Europhys. Lett. 5, 437–442 (1988)] in the linear regime with the success of the Rolie-Poly constitutive equation [Likhtman et al., J. Non Newtonian Fluid Mech. 114, 1–12 (2003)] in the nonlinear regime. We first review the key concepts of the double reptation approximation and the original (monodisperse) Rolie-Poly constitutive model. Subsequently, we provide the details of our approximate models for the particular case of a bidisperse blend and show that these models naturally identify the effects from couplings between constraint release and chain retraction (i.e., the so-called “enhanced stretch relaxation time”). Finally, we generalize to a multicomponent (polydisperse) model, based on the same underlying principles. Along the way, we also show that both of our models are in qualitative, and largely quantitative, agreement with experimental data for bidisperse and polydisperse melts of entangled linear polymers.

Contour length fluctuations and constraint release in entangled polymers: Slip-spring simulations and their implications for binary blend rheology

We study the interaction between constraint release and contour length fluctuations in well entangled polymers, by means of a set of slip-spring simulations in which the rate of constraint release is precisely controlled. In the present simulations, a fraction f = 0.9 of the slip-springs undergo constraint release, while the remaining slip-springs do not, as an idealized model to mimic the constraint release environment of a bidisperse polymer melt. Both populations of slip-springs allow reptation of the chain. Our analysis of the data indicates the time and parameter regions in which contour length fluctuations occur effectively along a thin tube, or along a diluted fat tube. We predict the parameters for which case each is observed and the rate of relaxation due to contour length fluctuations in each region. Finally, we draw the implications of the simulations for bidisperse blend rheology, by revisiting the classic “Viovy diagram” for such melts. We relocate some of the lines on the original diagram, and identify new regimes, based on the physics and quantitative information supplied by the simulation data. In particular, we identify a new region in the diagram in which along-tube motion of the long chains is predominantly along the contour of the thin tube, yet contour length fluctuations occur in the fat tube, resulting in an acceleration of the terminal relaxation. We successfully and quantitatively locate a wide range of literature data on our redrawn diagram.

Linear viscoelastic behavior of bidisperse polystyrene blends: experiments and slip-link predictions

The linear viscoelastic behavior of three well-entangled linear monodisperse polystyrene melts and their blends is investigated. The monodisperse melts are blended in a 1:1 weight ratio to obtain three polystyrene bidisperse blends for which the linear viscoelastic behavior is also measured. Special attention is paid to controlling sample size and solvent content, and checking for consistency in the high-frequency regime. We also attempt to estimate uncertainty quantitatively. The experimental results agree well with the discrete slip-link model, a robust mesoscopic theory that has been successful in predicting the rheology of flexible entangled polymer liquids and gels. Using recently developed analytic expressions for the relaxation modulus, predictions of the monodisperse samples are made. The parameters for the model are obtained from the low-frequency crossover of one experiment. Using this parameter set without adjustment, predictions over the fully accessible experimental frequency range are obtained for the monodisperse samples and their blends with very good agreement.

Chain stretching and recoiling during startup and cessation of extensional flow of bidisperse polystyrene blends

Simultaneous measurements of stress and flow-induced chain alignment were collected on a series of bidisperse polystyrene blends during startup of uniaxial extension and after cessation of flow at 150 °C. Chain alignment was measured using a novel method that combines the use of a commercial Sentmanat extensional rheometer with time-resolved small-angle neutron scattering (SANS) measurements. The latter uses state of the art methods of neutron time stamping in the SANS detector and deconvolution protocols that yields scattering data with time resolutions of the order of seconds [M. A. Calabrese et al., Soft Matter 12, 2301–2308 (2016)]. Those measurements were used to confirm the direct correlation between strain hardening and chain stretching at strain rates corresponding to Rouse Weissenberg number WiR> 0.5. Furthermore, a linear relation between the alignment factor, Af, and the tensile stress, σE, was observed in the bidisperse blends for stress values below 65 kPa during flow startup. This result confirmed the validity of a simple stress-SANS rule (SSR), analogous to the stress-optic rule, which relates chain alignment to the extensional stress. For stresses below 65 kPa, the stress-SANS coefficient, CE=Af/σE, was found to have a value of 3.9 MPa−1. Failure of the SSR is observed at stresses greater than 65 kPa. Further confirmation of the SSR was provided by the linear relation between Af and σ during the late regime of relaxation after flow cessation, which yield CE values nearly identical to those obtained during flow startup.

Molecular dynamics simulations of monodisperse/bidisperse polymer melt crystallization

We use large scale coarse-grained molecular dynamics simulations to study the kinetics of polymer melt crystallization. For monodisperse polymer melts of several chain lengths under various cooling protocols, we show that short chains have a higher terminal crystallinity value compared to longer ones. They align at the early stages and then cease evolving. Long chains, however, align, fold into lamella structures and then slowly optimize their dangling ends for the remaining simulation time. We then identify the mechanism behind bidisperse blend crystallization. To this end, we introduce a new algorithm (called Individual Chain Crystallinity) that allows the calculation of the crystallinity separately for short and long chains in the blend. We find that, in general, bidispersity hinders crystallization significantly. At first the crystallinity of the long chain components exceeds that of the monodisperse melt, but subsequently falls below the corresponding monodisperse melt curve after a certain “crossover time.” The time of the crossover can be attributed to the time required for the full crystallization of the short chains. This indicates that at the early stages the short chains are helping long chains to crystallize. However, after all short chains have crystallized they start to hinder the crystallization of the long chains by obstructing their motion. Lastly, polymer crystallization upon various thermodynamic protocols is studied. Slower cooling is found to increase the crystallinity value. Upon an instantaneous deep quench and subsequent isothermal relaxation, the crystallinity grows rapidly with time at early stages and subsequently saturates. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2016

A hierarchical multimode molecular stress function model for linear polymer melts in extensional flows

A novel hierarchical multimode molecular stress function (HMMSF) model for linear polymer melts is proposed, which implements the basic ideas of (i) hierarchical relaxation, (ii) dynamic dilution, and (iii) interchain tube pressure. The capability of this approach is demonstrated in modeling the extensional viscosity data of monodisperse, bidisperse, and polydisperse linear polymer melts. Predictions of the HMMSF model, which are solely based on the linear-viscoelastic relaxation modulus and a single free model parameter, the segmental equilibration time, are compared to elongational viscosity data of monodisperse polystyrene melts and solutions as well as to the elongational viscosity data of a bidisperse blend of two monodisperse polystyrenes, and good agreement between model and experimental data is observed. By using a simplified relation between the Rouse stretch-relaxation times and the relaxation times of the melts, the modeling is extended to the uniaxial, equibiaxial, and planar extensional viscosity data of a high-density polyethylene, the uniaxial and equibiaxial extensional viscosity data of a polydisperse polystyrene, the elongational viscosity data of three high-density polyethylenes, and a linear low-density polyethylene. For polydisperse melts, the modeling is again based exclusively on the linear-viscoelastic relaxation modulus with only one material parameter, the dilution modulus, which quantifies the onset of dynamic dilution.

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.

The molecular stress function (MSF) model in rheology

The molecular stress function (MSF) model is an integral constitutive equation introduced more than two decades ago. It is based on the time-deformation separability principle. The time contribution encloses the linear viscoelastic information, which can be provided by the phenomenological models or any molecular theory. The deformation contribution is defined in the MSF model as a strain measure describing the orientation and the stretch of the strands of the chain as independent processes. The orientation is described by the second-order tensor of the Doi-Edwards model, considering the independent alignment assumption. The stretch is taken into account by the molecular stress function, the main characteristic being that it is included inside the integral and it is the solution of an evolution equation. Since its proposal, the MSF model has been used to describe quantitatively the non-linear rheology of a broad variety of materials such as rubbers, linear and long-chain branched polymer melts and blends of polydisperse samples relevant to the industry. Nearly, monodisperse systems in solution and melt states have also been studied in samples with different structures like linear, bidisperse blends with linear components, combs and pom-pom molecules. Predictions have been obtained for a variety of deformations like uniaxial, equibiaxial and planar extensional flow as well as for steady, medium and large amplitude oscillatory and exponential shear flow. The quantitative description of polymer melts in transient elongation is crucial for numerical simulations. Therefore, the MSF model has been applied to perform finite element simulations for different processes and freesurface deformations, due to its flexibility, reliability and reduced number of material parameters. The integral constitutive equation and its physical interpretation remains the same since it was first published. The evolution equation of the molecular stress function is material dependent because it considers different molecular mechanisms occurring in different structures. Given its importance to rheology, it is the objective of this contribution to review the antecedents, physical basis and applications of the MSF model.

Extension-Induced Crystallization of Poly(ethylene oxide) Bidisperse Blends: An Entanglement Network Perspective

The role of long chains in extension flow-induced crystallization was studied with a combination of extension rheological and in situ small-angle X-ray scattering (SAXS) measurements at 52 °C. To elucidate the effects of long chains, bidisperse blends of poly(ethylene oxide) (PEO) with the long-chain concentration above the overlap concentration were prepared, constructing long-chain entanglement network in short-chain matrix. Rheological data of step extension on PEO melt are divided into two regions with fracture strain of pure short-chain sample as a boundary. Distinctly different features of crystallization kinetics and crystal morphologies are observed in these two regions, exactly corresponding to rheological behavior. A new mechanism based on entanglement network perspective is proposed, in which the second entanglement network constructed by long chains has three effects: (i) helping flow to change the free energy of polymer melt more effectively; (ii) ensuring the specific work can impose on the ...

Effect of Matrix Polymer on Flow-Induced Nucleation in Polymer Blends

Long-chain molecules in polymers are responsible for flow-induced nucleation (FIN). The role of a short-chain matrix in FIN was investigated using bidisperse blends of polyethylenelike high molecular weight long chains in short-chain polymer matrices. It was found that the critical specific work, w(c), measured for the onset of FIN in the blends has a power law dependence on the molecular weight of the matrix polymer with w(c) proportional M(w)(2.59 ± 0.27) indicating that the matrix can affect the stretching of the long chains and limit their ability to act as nucleation sites for flow-induced crystallization.

Ring Conformations in Bidisperse Blends of Ring Polymers

The size of rings (also called cyclic polymers) in bidisperse blends of chemically identical rings is analyzed by computer simulations. Data of entangled ring blends and blends of interpenetrating rings are compared and it is shown that the compression of entangled rings can be explained by the changes in the penetrable fraction of the minimal surface bounded by the ring. Corrections for small rings can be approximated by a concatenation probability $1-P_{OO}$ that a ring entraps at least one other ring. Both results are in line with a previous work [1] to explain the compression of entangled rings in monodisperse melts. Bond-bond correlations in melts of interpenetrating rings lead to similar corrections for ring sizes as reported previously [2] for monodisperse linear melts. For entangled rings, bond-bond correlations show an anti-correlation peak at a curvilinear distance of about ten segments that coincides with a horizontal tangent in the normalized mean square internal distances along the ring. Both observations become independent of melt molecular weight for sufficiently large degrees of polymerization and such behavior is not found in samples with entanglements switched off. In consequence, the length scale of topological interactions (entanglement length) in a melt of entangled rings must be considered as constant in contrast to a recent proposal by Sakaue [3].

A full-chain constitutive model for bidisperse blends of linear polymers

We develop a full-chain tube-based constitutive model [along the lines of Graham et al. J. Rheol. 47, 1171 (2003)] for the nonlinear rheology of bidisperse blends of long and short linear polymers. For a test chain in the blend, we use the physical picture of a fat tube, representing long-lived entanglements with long chains, and a thin tube, representing entanglements with all chains. The model includes the processes of reptation, contour length fluctuation (CLF), constraint release, and stretch relaxation. In the linear rheology regime, we identify a new relaxation process: CLF along the fat tube contour, achieved via a combination of chain motion along the thin tube, and local constraint release of the thin tube as it explores the width of the fat tube. This process is sufficiently fast to relax a significant portion of the long chains before reptation. It provides an explanation of the decrease in terminal time of long chains upon dilution with short chains in a framework where motion along the thin tube is the dominant reptation mechanism. Once the linear rheology is matched, nonlinear rheology is predicted with no further adjustments to the model. The model compares well against several experimental datasets on bidisperse blends. In particular, it predicts the onset rate of extension hardening, which is often significantly below the inverse Rouse time of the long chains.

Basic characteristics of uniaxial extension rheology: Comparing monodisperse and bidisperse polymer melts

We have carried out continuous and step uniaxial extension experiments on monodisperse and bidisperse styrene-butadiene random copolymers (SBR) to demonstrate that their nonlinear rheological behavior can be understood in terms of yielding through breakdown of the chain entanglement network and rubberlike rupture via non-Gaussian chain stretching leading to chain scission, respectively. In continuous extension, the sample with bidisperse molecular weight distribution showed greater resistance, due to double-networking, against the yielding-initiated failure. An introduction of 20% high molecular weight (106 g/mol) SBR to an SBR matrix (2.4 × 105 g/mol) could postpone the onset of nonuniform extension by as much as two Hencky strain units. In step extension, the bidisperse blends were also found to be more resistant to elastic breakup than the monodisperse matrix SBR. Rupture in both monodisperse and bidisperse SBR samples occurred when the finite chain extensibility was reached at sufficiently high rates. It is important to point out here that these results along with the concept of yielding allow us to clarify the concept of strain hardening in extensional rheology of entangled polymers.

Ellipsometry as a probe of crystallization in binary blends of a sphere-forming diblock copolymer

Ellipsometry is used to measure the crystallization and melting temperature of a bidisperse blend of a crystalline-amorphous diblock copolymer. Binary blends of sphere-forming poly(butadiene-ethylene oxide) (PB-PEO) of two different molecular weights are prepared. The two PB-PEO diblocks that are used share the same amorphous majority PB block length but different crystalline PEO minority block length. As the concentration of higher molecular weight diblock in the blend is increased, the size of the PEO spherical domains swell, providing access to the full range of domain sizes between the limits of the two neat diblock components. The change in domain size is consistent with a monotonic change in both the crystallization and melting temperatures. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2011