Modeling elongational viscosity of blends of linear and long-chain branched polypropylenes

To enhance the melt strength of a conventional linear polypropylene (L-PP), blends with a long-chain branched polypropylene (LCB-PP) were produced by adding 2, 5, 10, 25, 50, and 75 wt% of LCB-PP to L-PP and mixing in a twin screw extruder. It was found that, already, an addition of 10% or less of LCB-PP to L-PP leads to significant strain hardening. Elongational viscosity data of L-PP and LCB-PP and those of their blends were analyzed by the use of the molecular stress function (MSF) theory. While L-PP is characterized by the MSF parameter, β=1 (typical for linear melts), and shows very little chain stretch (), melt elongational behavior of LCB-PP is characterized by the MSF parameters, β=2 (typical of LCB melts), and (which corresponds to a maximum stretch of molecular chains by a factor of 15). By extruding LCB-PP, a refining effect is observed similar to the refining effects seen in low density polyethylene (LDPE), which reduces the steady-state elongational viscosity and reduces to 121. A second-order mixing rule for the fractional relaxation moduli, g _i , was found to show good agreement with the linear-viscoelastic data of the blends. To simulate the elongational viscosities of the L-PP/LCB-PP blends, a similar second-order mixing rule was used for the MSF parameter, β, while a first-order mixing rule was found to be appropriate for . This allows for a quantitative prediction of the time-dependent elongational viscosities of all L-PP/LCB-PP blends on the basis of the linear and nonlinear parameters of the mixing components L-PP and LCB-PP only. Comparison between the steady-state elongational viscosities as obtained from creep experiments shows good agreement with predictions.     

Uniaxial elongational viscosity of PS/a small amount of UHMW-PS blends

A series of polystyrene (PS) and a small amount of ultra high molecular weight (UHMW) PS blends have been prepared by using tetrahydrofuran (THF). Matrix PS has an M_w of 423,000 (M_w/M_n= 2.36) and UHMW-PS has either an M_w of 3,220,000 (M_w/M_n= 1.05) or 15,400,000 (M_w/M_n=1.30) in the range of concentration from 0 wt% to 1.5 wt%. The influence of a small amount of UHMW on dynamic viscoelasticity was investigated. At the frequency lower than 0.001 rad/s, the enhancement of G′ was observed by the incorporation of a small amount of UHMW. And the degree of enhancement was in the order of M_w of UHMW and its concentration. The measurement of uniaxial elongational viscosity for the blends was performed and the effects of UHMW on strain-hardening properties were analyzed at equal strain-rate conditions. The concentration of UHMW where the strain-hardening becomes substantially stronger was determined. To get more insight into the cause of enhancement of strain-hardening at a certain concentration, the damping function from step-shear stress relaxation was measured. The influence of a small amount of UHMW on the damping function was found to be small. It was interpreted, from time- and strain-dependency points, that the enhancement of strain-hardening by a small amount of UHMW was governed by the long relaxation time. Received: 6 September 2000 Accepted: 11 January 2001     

Determination of elongational viscosity of polymer melts by RME and Rheotens experiments

Several linear (LLDPE, HDPE, PS) and long-chain-branched (LDPE, PP) polymer melts were investigated by an elongational rheometer (RME Rheometrics) and by Rheotens (Göttfert). The Molecular Stress Function (MSF) theory is briefly reviewed and used to extrapolate the steady-state elongational viscosity. To evaluate Rheotens experiments, a new process model is introduced which assumes that the elongational viscosity in the Rheotens test is a function of the draw ratio only. The apparent elongational viscosities extracted from Rheotens curves are found to lie in between the steady-state elongational viscosity and three times the shear viscosity.     

Recoverable deformation and morphology after uniaxial elongation of a polystyrene/linear low density polyethylene blend

The transient recoverable deformation ratio after melt elongation at various elongational rates and maximum elongations was investigated for pure polystyrene and for a 85 wt.% polystyrene/15 wt.% linear low density polyethylene (PS/LLDPE 85:15) blend at a temperature of 170 ^oC. The ratio p of the zero shear rate viscosity of LLDPE to that of PS is p = 0.059 ≈ 1:17. Retraction of the elongated LLDPE droplets back to spheres and end-pinching is observed during recovery. A simple additive rule is applied in order to extract the contribution of the recovery of the elongated droplets from the total recovery of the blend. In that way, the recoverable portion of the PS/LLDPE blend induced by the interfacial tension is determined and compared with the results of a theory based on an effective medium approximation. The effective medium approximation reproduces well the time scale of the experimental data. In addition, the trends that the recoverable deformation increases with elongational rate and maximum elongation are captured by the theoretical approach.     

Linear viscoelasticity,simple and planar melt extension of linear polybutadienes with bimodal molar mass distributions

Shear oscillations, simple and planar elongations have been performed with anionically polymerized polybutadienes (PB) and their blends at room temperature. The PB components were of different molar mass averages and of narrow molar mass distributions; the blends had bimodal molar mass distributions and are represented by the weight ratio w of the high molecular component. The crossover G() = G() obtained from oscillatory measurements shows correlations with molecular parameters. For the zero shear viscosity the well-known relation ₀ M _w ^3.4 is found. The recoverable equilibrium shear compliance J _e ⁰ is nearly the same for the components; for the blends it strongly depends on w with a pronounced maximum at small w. In elongation outside the linear region strain hardening is found; its magnitude depends on M _w of the components, the composition w of the blend, the mode of elongation (simple or planar), and the elongational strain rate. The hardening revealed in the increase of the elongational viscosity above the linear viscoelastic limit increases as a function of w up to a maximum similar to J _e ⁰ such that, for both properties, the molecular processes may be the same. The elongational viscosity µ₂ (from the lateral stress in planar elongation) is above the linear viscoelastic limit for bimodal and below this limit for conventional broad molar mass distributions. In general, it can be stated that with a more narrow molar mass distribution of linear polymers the elongational behavior of the melts comes closer to the linear viscoelastic limit.Dedicated to Professor Arthur S. Lodge on the occasion of his 70th birthday and his retirement from the University of Wisconsin.Extended version of a paper presented at the Annual Conf. German Soc. of Rheology, Berlin, May 13–15, 1991.     

The influence of blending polystyrenes of narrow molecular weight distribution on melt creep flow and creep recovery in elongation

Creep and creep recovery experiments in elongation were performed with melts of anionically polymerized polystyrenes (PS) and with their blends at a temperature of 150 °C. For stresses ₀ < 10 000 N/m² the samples with narrow molecular weight distribution show linear viscoelastic behavior up to the maximum Hencky strain = 3.5, achievable in a newly developed elongational rheometer for polymer melts. The compliances,D (t), of the blends are linear-viscoelastic only up to a strain limit _L . For strains beyond _L the compliance of each blend depends on the stress ₀. For a series of binary blends, prepared from the same components of narrow MWD, the linear-viscoelastic limit _L seems to be independent of the mixing ratio and stress. _L seems to be a function only of the molecular weights of the original components, the blends investigated were made from.Paper presented at the Annual Conference of the German Society of Rheology at Ulm, March 7–10, 1983.     

Large amplitude oscillatory elongation flow

A filament stretching rheometer (FSR) was used for measuring the elongation flow with a large amplitude oscillative elongation imposed upon the flow. The large amplitude oscillation imposed upon the elongational flow as a function of the time t was defined as where ε is the Hencky strain, is a constant elongational rate for the base elongational flow, Λ the strain amplitude (Λ ≥ 0), and Ω the strain frequency. A narrow molecular mass distribution linear polystyrene with a molecular weight of 145 kg/mol was subjected to the oscillative flow. The onset of the steady periodic regime is reached at the same Hencky strain as the onset of the steady elongational viscosity ( Λ = 0). The integral molecular stress function formulation within the ‘interchain pressure’ concept agrees qualitatively with the experiments.     

Exponential shear flow of branched polyethylenes in rotational parallel-plate geometry

Exponential shear flow, as a strong flow with the potential to generate a high degree of molecular stretching, has attracted considerable interest in recent years. So far, exponential shear flow has been realized by either sliding-plate or cone-and-plate (CP) geometry. Both geometries guarantee homogeneous shear flow. Here, we present experimental data on exponential shear flow of several long-chain branched polyethylene melts with different degrees of strain hardening obtained by using parallel-plate (PP) geometry in a rotational rheometer. This type of geometry, which is standard in linear-viscoelastic characterization of polymer materials, produces inhomogeneous shear flow. A comparison of exponential shear flow data obtained by PP and CP geometry is made. Additionally, the experimental data are compared to predictions of the rubber-like liquid (RLL) and the molecular stress function (MSF) theories. For this purpose, the relaxation spectra of the polymer melts considered were obtained by standard linear-viscoelastic characterization. In addition, two irrotational parameters and one rotational parameter are required by the MSF theory. While the irrotational parameters were obtained from fitting to elongational viscosity data, the value of the rotational parameter was used as given in the literature. It can be concluded that viable experimental data in exponential shear flow can be obtained by PP geometry. For finite linear-viscoelasticity (RLL theory), predictions of reduced shear stress for CP and PP geometry coincide, but nonlinear material behavior (as modeled by the MSF theory) leads to small differences between both geometries. Furthermore, it is shown that the MSF predictions are in excellent agreement with the experimental data in exponential shear flow and that this type of flow leads to much less chain stretching than elongational flow.Dedicated to the memory of Prof. Arthur S. Lodge (1922–2005).     

A modification of the convective constraint release mechanism in the molecular stress function model giving enhanced vortex growth

The molecular stress function model with convective constraint release (MSF with CCR) constitutive model [M.H. Wagner, P. Rubio, H. Bastian, The molecular stress function model for polydisperse polymer melts with dissipative convective constraint release, J. Rheol. 45 (2001) 1387] is capable of fitting all viscometric data for IUPAC LDPE, with only two adjustable parameters (with difference found only on reported “steady-state” elongational viscosities). The full MSF with CCR model is implemented in a backwards particle-tracking implementation, using an adaptive method for the computation of relative stretch that reduces simulation time many-fold, with insignificant loss of accuracy. The model is shown to give improved results over earlier versions of the MSF (without CCR) when compared to well-known experimental data from White and Kondo [J.L. White, A. Kondo, Flow patterns in polyethylene and polystyrene melts during extrusion through a die entry region: measurement and interpretation, J. Non-Newtonian Fluid Mech. 3 (1977) 41]; but still to under-predict contraction flow opening angles. The discrepancy is traced to the interaction between the rotational dissipative function and the large stretch levels caused by the contraction flow. A modified combination of dissipative functions in the constraint release mechanism is proposed, which aims to reduce this interaction to allow greater strain hardening in a mixed flow. The modified constraint release mechanism is shown to fit viscometric rheological data equally well, but to give opening angles in the complex contraction flow that are much closer to the experimental data from White and Kondo. It is shown (we believe for the first time) that a constitutive model demonstrates an accurate fit to all planar elongational, uniaxial elongational and shear viscometric data, with a simultaneous agreement with this well-known experimental opening angle data. The sensitivity of results to inaccuracies caused by representing the components of the deformation gradient tensor to finite precision is examined; results are found to be insensitive to even large reductions in the precision used for the representation of components. It is shown that two models that give identical response in elongational flow, and a very similar fit to available shear data, give significantly different results in flows containing a mix of deformation modes. The implication for constitutive models is that evaluation against mixed deformation mode flow data is desirable in addition to evaluation against viscometric measurements.     

Viscoelastic properties of POSS–styrene nanocomposite blended with polystyrene

Polyhedral oligomeric silsesquioxane (POSS) are hybrid nanostructures of about 1.5 nm in size. These silicon (Si)-based polyhedral nanostructures are attached to a polystyrene (PS) backbone to produce a polymer nanocomposite (POSS–styrene). We have solution blended POSS–styrene of with commercial polystyrene (PS), , and studied the rheological behavior and thermal properties of the neat polymeric components and their blends. The concentration of POSS–styrene was varied from 3 up to 20 wt.%. Thermal analysis studies suggest phase miscibility between POSS–styrene and the PS matrix. The blends displayed linear viscoelastic regime and the time–temperature superposition principle applied to all blends. The flow activation energy of the blends decreased gradually with respect to the matrix as the POSS–styrene concentration increased. Strikingly, it was found that POSS–styrene promoted a monotonic decrease of zero-shear rate viscosity, η ₀, as the concentration increased. Rheological data analyses showed that the POSS–styrene increased the fractional free volume and decreased the entanglement molecular weight in the blends. In contrast, blending the commercial PS with a PS of did not show the same lubrication effect as POSS–styrene. Therefore, it is suggested that POSS particles are responsible for the monotonic reduction of zero-shear rate viscosity in the blends.     

Effect of molecular weight and shear on phase diagram of PS/PVME blend

The molecular weight dependence of the lower critical solution temperature of polystyrene (PS) and poly(vinyl methyl ether) blends was studied by laser light transmission. The temperature of phase separation was found to decrease with increasing PS molecular weight. In the steady shear flow conditions and near the critical temperature, shear was found to enhance fluctuations of concentration at relatively small shear rates, whereas it suppresses such fluctuations at high shear rates. The shift in Flory-interaction parameter Δ_χ was calculated from experimental data and its sign was used to predict shear-induced mixing or shear-induced demixing under flow field. The obtained experimental results were compared to Criado-Sancho et al. and Clarke-Mcleish models.     

Uniaxial deformation and relaxation of polymer blends: relationship between flow and morphology development

Uniaxial elongational flow followed by stress relaxation of a dilute mixture of polystyrene/polymethylmethacrylate) PS/PMMA with PS (5 wt%) as a dispersed phase was investigated. The behavior of the blend was found to be dominated by the PMMA matrix during elongation and by the interface during the relaxation at long time. Such a behavior was related to drop deformation and shape recovery during the relaxation process as was confirmed by morphological analyses on samples quenched within the rheometer just after elongation and at various times during the relaxation process. The morphology and the rheological material functions variation were compared to the Yu model (Yu W, Bousmina M, Grmela M, Palierne JF, Zhou C (2002) Quantitative relationship between rheology and morphology in emulsions. J Rheol 46(6):1381–1399).     

Nonlinear viscoelasticity of PP/PS/SEBS blends

The nonlinear viscoelastic behavior of polypropylene/polystyrene (PP/PS) blends compatibilized or not with the linear triblock copolymer (styrene-ethylene-/butylene-styrene, SEBS) was investigated. Start-up of steady-shear at rates from 0.1 to 10 s^–1 was carried out using a controlled strain rotational rheometer and a sliding plate rheometer for strain histories involving one or several shear rates. The shear stress and first normal shear stress difference were measured as functions of time, and the morphologies of the samples before and after shearing were determined. For each strain history except that involving a single shear rate of 0.1 s^–1 the blends showed typical non-linear viscoelastic behavior: a shear stress overshoot/undershoot, depending on the history, followed by a steady state for each step. The first normal stress difference increased monotonically to a steady-state value. The values of the stresses increased with the addition of SEBS. The shear stress overshoot and undershoot and the times at which they occurred depended strongly on the strain history, decreasing for a subsequent shear rate step performed in the same direction as the former, and the time at which stress undershoot occurred increased for a subsequent shear rate step performed in the opposite direction, irrespective of the magnitude of the shear rate. This behavior was observed for all the blends studied. The time of overshoot in a single-step shear rate experiment is inversely proportional to the shear rate, and the steady-state value of N₁ scaled linearly with shear rate, whereas the steady-state shear stress did not. The average diameter of the dispersed phase decreased for all strain histories when the blend was not compatibilized. When the blend was compatibilized, the average diameter of the dispersed phase changed only during the stronger flows. Experimental data were compared with the predictions of a model formulated using ideas of Doi and Ohta (1991), Lacroix et al. (1998) and Bousmina et al. (2001). The model correctly predicted the behavior of the uncompatibilized blends for single-step shear rates but not that of the compatibilized blends, nor did it predict morphologies after shearing.     

The interchain pressure effect in shear rheology

Recently, the tube diameter relaxation time in the evolution equation of the molecular stress function (MSF) model (Wagner et al., J Rheol 49: 1317–1327, 2005) with the interchain pressure effect (Marrucci and Ianniruberto, Macromolecules 37:3934–3942, 2004) included was shown to be equal to three times the Rouse time in the limit of small chain stretch. From this result, an advanced version of the MSF model was proposed, allowing modeling of the transient and steady-state elongational viscosity data of monodisperse polystyrene melts without using any nonlinear parameter, i.e., solely based on the linear viscoelastic characterization of the melts (Wagner and Rolón-Garrido 2009a, b). In this work, the same approach is extended to model experimental data in shear flow. The shear viscosity of two polybutadiene solutions (Ravindranath and Wang, J Rheol 52(3):681–695, 2008), of four styrene-butadiene random copolymer melts (Boukany et al., J Rheol 53(3):617–629, 2009), and of four polyisoprene melts (Auhl et al., J Rheol 52(3):801–835, 2008) as well as the shear viscosity and the first and second normal stress differences of a polystyrene melt (Schweizer et al., J Rheol 48(6):1345–1363, 2004), are analyzed. The capability of the MSF model with the interchain pressure effect included in the evolution equation of the chain stretch to model shear rheology on the basis of linear viscoelastic data alone is confirmed.     

The role of short chain molecules for the rheology of polystyrene melts.

Atactic polystyrenes of narrow molar mass distribution with average molar masses larger than the critical molar massM _c were mixed with similar polystyrenes of molecular masses lower thanM _c . Linear viscoelastic melt properties of these binary blends were measured with a dynamic viscometer of the concentric cylinder type. One of the experimental findings is that the time-temperature shift factorsa _T are dependent on the composition of the samples. This can be understood, if free volume due to chain-ends is taken into account. A computer-fitted WLF-equation being modified in a proper way leads to the following results: At the glass-transition-temperature the fraction of free volume in polystyrene of infinite molar mass is only 0.015. At a temperature of 180 °C the mean value of the free volume at a chain end is 0.029 nm³ for the polystyrene investigated.     

Rheological characterization of cross-linked poly(methyl methacrylate)

Poly(methyl methacrylate) (PMMA) with various degrees of cross-linking were prepared from methyl methacrylate and a cross-linker, and the effect of dilution of the polymerizable mixture by a thermoplastic PMMA on the cross-linked PMMAs was evaluated. The rheological properties were characterized in linear viscoelasticity and in uniaxial extensional flow. A critical gel is formed at concentrations of the cross-linking agent neopentyl glycol dimethacrylate (NPG) of approximately 250 mol ppm both in the case of PMMAs, which are not diluted by an addition of thermoplastic PMMA to the monomer (Recipe-A), and of PMMAs, which were obtained by an addition of 25 wt% low molecular weight thermoplastic PMMA to the monomer (Recipe-B). Significant strain hardening is observed for concentrations of NPG at and above 100 mol ppm for PMMAs based on Recipe-A and for all PMMAs produced by Recipe-B. At the same NPG concentration of 30 mol ppm, PMMA produced by Recipe-A shows very little strain hardening, while PMMA produced by Recipe-B shows significant strain hardening. This is due to the difference in the molecular weight distribution: PMMA from Recipe-A is mono-modal with M _w /M _n = 2.5, while PMMA from Recipe-B is bimodal with M _w /M _n = 5.6. Surprisingly, the strain-hardening tendency is strongly increasing with increasing NPG concentration, and at the same NPG concentration, the strain hardening of PMMAs produced by Recipe-B is higher than that of PMMAs produced by Recipe-A. This difference can be attributed to the dilution effect of the (unreacted) thermoplastic PMMA in Recipe-B PMMAs. The elongational flow behavior was also analyzed by the Molecular Stress Function (MSF) model.     

Rheo-optical measurements of the elongational flow of aqueous polymer solutions

An opposed jets rheometer was used to investigate the elongational viscosity as a function of the strain rate for dilute aqueous solutions of polyvinylformamide and polyacrylamide. Critical strain rates at which the enhancement in elongational viscosity occurs were determined for both systems. The influence of the polymer concentration on the elongational viscosity was investigated. The measurements were performed with polymer concentrations less than the critical polymer concentration c_p^* c_p^* . In order to assess the deformation and orientation of the macromolecules, flow-induced birefringence was measured simultaneously.     

The effect of 1,3:2,4-<Emphasis Type="Italic">bis</Emphasis>-<Emphasis Type="Italic">O</Emphasis>-(<Emphasis Type="Italic">p</Emphasis>-methylbenzylidene)-<Emphasis Type="SmallCaps">d</Emphasis>-sorbitol (PDTS) on uniaxial elongational viscosity of polypropylene

We investigated the dynamic viscoelasticity and elongational viscosity of polypropylene (PP) containing 0.5 wt% of 1,3:2,4-bis-O-(p-methylbenzylidene)-d-sorbitol (PDTS). The PP/PDTS system exhibited a sol–gel transition (T _gel) at 193 °C. The critical exponent n was nearly equal to 2/3, in agreement with the value predicted by a percolation theory. This critical gel is due to a three-dimensional network structure of PDTS crystals. The elongational viscosity behavior of neat PP followed the linear viscosity growth function 3η ⁺ (t), where η ⁺ (t) is the shear stress growth function in the linear viscoelastic region. The elongational viscosity of the PP/PDTS system also followed the 3η ⁺ (t) above T _gel but did not follow the 3η ⁺ (t) and exhibited strong strain-softening behavior below T _gel. This strain softening can be attributed to breakage of the network structure of PDTS with a critical stress (σ _c) of about 10⁴ Pa.