Effects of chain length distribution on viscoelastic properties of entangled linear polymers: Blending laws for binary blends

The reptation idea of de Gennes and the tube model theory of Doi and Edwards are extended to explain the terminal viscoelastic properties of binary blends in the highly entangled state of two linear monodisperse polymers with different molecular weights M₁ and M₂. A modified tube model is proposed that considers the significance of the constraint release by local tube renewal in accounting for the relaxation process of the higher molecular weight chain. Its relaxation by both reptation and the constraint release is remodeled as the disengagement by pure reptation of an equivalent primitive chain. From knowledge of the longest relaxation times of the blend components, the stress equation is formulated from which blending laws of viscoelastic properties for the binary blends are derived. To force better agreement between theory and experiment at the pure monodisperse limits of the blends, a crude treatment to include the effect of contour-length fluctuation in the equivalent-chain model is also discussed. Theoretical predictions of the zero-shear viscosity and steady-state shear compliance are shown to be in good agreement with literature data on undiluted polystyrenes and polybutadienes over a wide range of the blend composition and M₂/M₁ ratio. The asymptotic of the laws for blends with M₂/M₁ → 1 and 0 are comparable to those from the relaxation spectrum proposed by others earlier on the basis of the tube model.     

Viscoelastic properties of polymers: From molecular models to non-linear behavior

The molecular models of polymer physics (reptation, tube renewal) give a reasonable picture of the diffusion and relaxation of long and flexible chains: the concept of “tube renewal” (constraint release) added to the reptation idea explains the polydispersity effects for multimodal blends as well as for commercial linear polymers. The real issue now is to introduce these concepts in the formalism of non-linear viscoelasticity in order to explain the experimental data, as a first step in the range of moderate rates of deformation, then at very high strains.     

Universality of linear viscoelasticity of monodisperse linear polymers

We suggest a universal plot that superposes linear viscoelastic data of nearly monodisperse polymers on a single curve, regardless of the molecular weight, temperature, and species of polymers. The plotting method is based on the time–temperature superposition and rescaling of viscoelastic functions with terminal behavior. Without any information from molecular theories, the plot supports the fact that the molecular theories of the linear viscoelasticity of monodisperse polymers are independent of the species of polymers. Although an appropriate scaling may show universality by separately extracting the reptational mode and the Rouse mode from the whole set of viscoelastic data, our plotting method shows universality in a unified manner that scales the viscoelastic functions measured over the whole frequency range. We explain the origin of the universality of the plot in terms of molecular theory, the phenomenological spectra of the relaxation time (the BSW and CW spectra), and the principle of time–temperature superposition. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 2730–2737, 2004     

Tube dilation process in star‐branched cis‐polyisoprenes

In current tube models for entanglement, the tube representing the topological constraint is considered to move with time. This tube motion results in the constraint release (CR) as well as the dynamic tube dilation (DTD), and an importance of DTD has been argued for entangled star chains. Under these backgrounds, this article examines the validity of the DTD molecular picture for the star chains. For monodisperse star chains having noninverted type‐A (parallel) dipoles in respective arms, the normalized viscoelastic and dielectric relaxation functions μ(t) and Φ(t) were found to obey a relationship μ(t) ≅ [Φ(t)]² if the tube actually dilates in the time scale of the star relaxation. For 6‐arm star cis‐polyisoprene (PI) chains (having those type‐A dipoles), dielectric and viscoelastic measurements were conducted to test this DTD relationship. Both viscoelastic and dielectric properties exhibited characteristic behavior expected from DTD models (assuming the arm retraction in the dilating tube), the exponential increase of the relaxation time and broadening of the relaxation mode distribution with increasing arm molecular weight M_a. However, in the range of M_a examined, M_a ≤ 8M_e (M_e = entanglement spacing), the above DTD relationship was not valid for a dominant part of the slow relaxation (and the models failed in this sense). Thus, for star chains at least in this range of M_a, the simple DTD picture assuming very rapid CR motion (rapid equilibration in the dilated tube) did not explain the slow relaxation behavior of star chains. This result in turn suggested the importance of the CR motion in this behavior. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 1024–1036, 2000     

Molecular tube models of branched polymer melts

We apply the tube model to a specific problem in polymer melt dynamics - the rheology of star polymers as an additive to a monodisperse linear matrix. We find that the tube dilation picture of constraint release may be applied to the relaxation of the star fraction. There are four qualitatively different cases depending on the relative concentrations and relaxation times of the two fractions. Terminal relaxation times and relaxation spectra are calculated for each case and compared with available experimental data. The existence of a modified Rouse relaxation such that G(t)∼t^−3/2 is argued.     

Rheology of polymer melts and concentrated solutions

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.     

Bimolecular radical termination: New perspectives and insights

The reversible addition‐fragmentation chain transfer‐chain length dependent termination (RAFT‐CLD‐T) method has allowed us to answer a number of fundamental questions regarding the mechanism of diffusion‐controlled bimolecular termination in free‐radical polymerization (FRP). We carried out RAFT‐mediated polymerizations of methyl acrylate (MA) in the presence of a star matrix to develop an understanding of the effect of polymer matrix architecture on the termination of linear polyMA radicals and compared this to polystyrene, polymethyl methacrylate, and polyvinyl acetate systems. It was found that the matrix architecture had little or no influence on termination in the dilute regime. However, due to the smaller hydrodynamic volumes of the stars in solution compared to linear polymer of the same molecular weight, the gel onset point occurred at greater conversions, and supported the postulate that chain overlap (or c*) is the main cause for the observed autoacceleration observed in FRP. Other theories based on “short–long” termination or free‐volume should be disregarded. Additionally, since our systems are well below the entanglement molecular weight, entanglements should also be disregarded as the cause of the gel onset. The semidilute regime occurs over a small conversion range and is difficult to quantify. However, we obtain accurate dependencies for termination in the concentrated regime, and observed that the star polymers (through the tethering of the arms) provided constriction points in the matrix that significantly slow the diffusion of linear polymeric radicals. Although, this could at first sight be postulated to be due to reptation, the dependencies showed that reptation could be considered only at very high conversions (close to the glass transition regime). In general, we find from our data that the polymer matrix is much more mobile than what is expected if reptation were to dominate. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 3155–3173, 2008     

Nonlinear stress relaxation of an entangled linear polystyrene in single step‐strain flow: A quantitative theoretical investigation

The nonlinear stress relaxation of a nearly monodisperse, moderately entangled polystyrene solution (i.e., roughly seven entanglements per chain at equilibrium) in single step‐strain flow is investigated quantitatively by a detailed comparison of an existing set of experimental data with a simulation based on the tube model. The proposed simulation enables the effects of primary nonlinear relaxation mechanisms other than chain retraction to be identified more clearly and investigated individually. Two peculiar nonlinear relaxation behaviors are observed in this experiment. One is concerned with an apparent enhancement in the stress relaxation at short times, and the other is responsible for a seeming slowdown of the stress relaxation at long times. These findings are discussed within the tube model, in view of the effects of convective constraint release, partial strand extension, and nonaffine deformation. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1281–1293, 2003     

Star polystyrenes by anionic star–star coupling reactions with divinylbenzene

The incremental addition of divinylbenzene was used to generate star polystyrenes in a nearly full conversion of a living polystyrenyl anion. The dramatic increase in the molecular weight of the star polystyrenes with a limited supply of the living polystyrene supported the formation of gradient‐star polystyrenes through star–star coupling. The stoichiometric analysis of the star polymers revealed that their connection polymer had a shorter length than their branch polymer. The measured solution viscosity of the gradient‐star polymers greatly deviated from a linear correlation with the molecular weight and was in parallel to a theoretical simulation based on a highly branched structure of the gradient‐star polymer. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 2579–2586, 2006     

Prediction and estimation of solvent diffusivities in polyacrylate and polymethacrylates

The relationship between polymer side‐chain length and the hole free volume that is effective for solvent diffusion was investigated for polyacrylates and polymethacrylates on the basis of free‐volume theory. Measurements of a polymer's viscoelasticity and solvent diffusivity provided experimental evidence for polymer segment mobility, and the results indicated that hole free volume in a linear polymer increases with hydrocarbon side‐chain length. Because the molecular mechanisms of polymer viscoelasticity and diffusivity are identical, the free‐volume parameters obtained for polyacrylates and polymethacrylates by measuring the polymer viscoelastic‐temperature dependence can reliably be used in predicting the solvent diffusion coefficient. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1393–1400, 2003     

Short‐time stretch relaxation of entangled polymer solutions investigated using full Rouse model predictions

We conduct a systematical investigation into the short‐time stretch relaxation behavior (i.e., shorter than the Rouse time but sufficiently longer than the glassy time) of entangled polymer liquid in single‐step strain flows, on the basis of theory/data comparisons for a broad series of type‐A entangled polymer solutions. First, within existing normal‐mode formulations, the Rouse model predictions on a full‐chain stretch relaxation in single‐step strain flows are derived for a popular 1‐D model proposed within the Doi–Edwards tube model, as well as for the original 3‐D model for nonentangled systems. In addition, an existing formula for the aforementioned 1‐D model that, however, rested upon a consistent‐averaging or the so‐called uniform‐chain‐stretch approximation is simultaneously examined. Subsequently, the previously derived formulas on chain stretch relaxation are directly incorporated into a reliable mean‐field tube model that utilizes the linear relaxation spectrum and the Rouse time constant consistently determined from linear viscoelastic data. It is found that the predictions of the 1‐D model differ substantially from that of the original 3‐D model at short times. Theory/data comparisons further indicate that the 1‐D model without approximations seems able to describe fairly well the nonlinear relaxation data under investigation. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1199–1211, 2006     

Highly entangled polymer primitive chain network simulations based on dynamic tube dilation

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.     

Using the new in–in method for the synthesis of polyisoprene/polystyrene heteroarm star polymers

The basis of the two‐step in–in method is as follows: star polymers with poly(divinyl benzene) cores, synthesized by the arm‐first method, include many unreacted double bonds in their core, and these double bonds can be attacked by the carbanions of some monomers such as styrene and dienes. In this work, linear polyisoprene chains were used to attack the double bonds existing in the poly(divinyl benzene) cores of polystyrene star polymers, so that a heteroarm star polymer with polystyrene and polyisoprene arms was synthesized. It was later well characterized with size exclusion chromatography, light scattering, viscometry, UV spectroscopy, dynamic mechanical thermal analysis, and ¹H NMR. © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 135–142, 2003     

Synthesis and characterization of 4‐arm star side‐chain liquid crystalline polymers containing azobenzene with different terminal substituents via ATRP

4‐Arm star side‐chain liquid crystalline (LC) polymers containing azobenzene with different terminal substituents were synthesized by atom transfer radical polymerization (ATRP). Tetrafunctional initiator prepared by the esterification between pentaerythritol and 2‐bromoisobutyryl bromide was utilized to initiate the polymerization of 6‐[4‐(4‐methoxyphenylazo)phenoxy]hexyl methacrylate (MMAzo) and 6‐[4‐(4‐ethoxyphenylazo)phenoxy]hexyl methacrylate (EMAzo), respectively. The 4‐arm star side‐chain LC polymer with p‐methoxyazobenzene moieties exhibits a smectic and a nematic phase, while that with p‐ethoxyazobenzene moieties shows only a nematic phase, which derives of different terminal substituents. The star polymers have similar LC behavior to the corresponding linear homopolymers, whereas transition temperatures decrease slightly. Both star polymers show photoresponsive isomerization under the irradiation with UV–vis light. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 3342–3348, 2007     

Difficult reptation of star‐shaped polystyrenes in their blends with poly(methyl methacrylate)

Films of polystyrene/poly(methyl methacrylate) (PS/PMMA) blends are obtained from solution after evaporation of the solvent. The degree of mixing of the two polymers is studied using scanning electron microscopy after selective elimination of the PS phase. Using star‐shaped instead of linear PS, an important degree of mixing is observed. This must be attributed to difficult reptation of the star‐shaped chains due to the high number of entanglements between star‐shaped PS and PMMA compared to the entanglements between linear PS and PMMA.     

Rheological properties of HyperMacs—long‐chain branched analogues of hyperbranched polymers

HyperMacs are long chain branched analogues of hyperbranched polymers, differing only in the sense that they have polymer chains, rather than monomers between branch points. Although the building blocks for HyperMacs and AB₂ macromonomers can be well defined in terms of molecular weight and polydispersity, the nature of the coupling strategy adopted for the synthesis of the HyperMacs results in branched polymers with a distribution of molecular weights and architectures. Melt rheology showed polystyrene HyperMacs to be thermorheologically simple, obeying William–Landel–Ferry behavior. Zero shear viscosities of the polymers were shown to increase with average molecular weight and the melts display shear‐thinning behavior. HyperMacs showed little evidence for relaxation by reptation and the rheological behavior agreed well with the Cayley tree model for hierarchical relaxation in tube models of branched polymers. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 2762–2769, 2007     

A toolbox for parameter-free predictions of solid-state properties of monodisperse glassy polymers with frozen-in molecular orientation

This study presents a toolbox for the prediction of birefringence and craze initiation stress in oriented monodisperse linear amorphous polymers. The toolbox is assembled from a previously proposed melt–solid constitutive model that provides the necessary residual stress components required for predictions of birefringence and craze initiation stress. The Likhtman–McLeish theory for linear rheology of entangled polymers is used to generate the low reduced frequency part of the linear viscoelastic spectrum, the only molar mass-dependent input parameter. All other parameters are obtained by experiment or from literature and can be considered material constants. Toolbox predictions are compared to new experimental data on two grades of linear monodisperse polystyrene (PS) of known molar mass but unknown rheology and to literature data. The toolbox is able to account for the role of molar mass on birefringence and craze initiation stress of PS subjected to supraentanglement orientation processes. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012     

Synthesis and postfunctionalization of well‐defined star polymers via “double” click chemistry

In this article, the synthesis and the functionalization of well‐defined, narrow polydispersity (polydispersity index < 1.2) star polymers via reversible addition‐fragmentation chain transfer polymerization is detailed. In this arm first approach, the initial synthesis of a poly(pentafluorophenyl acrylate) polymer, and subsequent, cross‐linking using bis‐acrylamide to prepare star polymers, has been achieved by reversible addition fragmentation chain transfer polymerization. These star polymers were functionalized using a variety of amino functional groups via nucleophilic substitution of pentafluorophenyl activated ester to yield star polymers with predesigned chemical functionality. This approach has allowed the synthesis of star glycopolymer using a very simple approach. Finally, the core of the stars was modified via thiol‐ene click chemistry reaction using fluorescein‐o‐acrylate and DyLigh 633 Maleimide. © 2011 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2011     

Local segmental relaxation in bidisperse polystyrenes

Dynamic mechanical results are reported for segmental relaxation of monodisperse polystyrenes (PSs) with molecular weights of 0.7, 3, 18, and 104 kg/mol and bidisperse PSs created from blending pairs of these materials. The data for the monodisperse polymers confirm previous findings; namely, there is an increase in the glass‐transition temperature normalized temperature dependence of the segmental relaxation times (fragility) with increasing molecular weight, along with a breakdown of the correlation between the fragility and the breadth of the relaxation function. For both the monodisperse and bidisperse PSs, the glass‐transition temperature is a single function of the average number of chain ends, independent of the nature of the molecular weight distribution. It is also found that these materials exhibit fragilities that uniquely depend on the number‐average molecular weight, that is, on the concentration of chain ends. In blends with linear PS, cyclic PS with a low molecular weight behaves as a high polymer, similar to its neat behavior, reflecting the overriding importance of chain ends. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 2604–2611, 2004     

Method of determining the absolute molecular weight from gel permeation chromatograms for star‐shaped styrenic block copolymers

A gel permeation chromatography (GPC) calculation method has been developed to determine the absolute molecular weight of a star‐shaped styrenic block copolymer with GPC–ultraviolet/refractive index calibrated with linear polystyrene standards. To illustrate the simplicity of this method, we have synthesized nearly monodisperse, multiple‐arm model polymers either by linking living polymeric arms with multifunctional silicon halide or by oligomerizing the p‐chloromethylstyrene‐terminated polystyrene macromonomers. The good agreement between the absolute molecular weight determined with this calculation method and that actually measured with a multi‐angle laser light scattering device has corroborated the validity of the calculation method. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 976–983, 2003