Dependence of the zero shear-rate viscosity and the viscosity function of linear high-density polyethylenes on the mass-average molar mass and polydispersity

Linear high-density polyethylenes with molar masses M _w between 240 and 1,000,000 g/mol, obtained by metallocene catalysts, were characterized in shear using oscillatory and creep tests. The polydispersities of the molar mass distributions (MMDs) lay between 1 and 16. The resulting zero shear-rate viscosities η₀ covered a range from 2.5×10⁻³ to around 10⁸ Pas. Above a critical molar mass of M _c≈2,900 g/mol, the experimental results can be described by the relation η₀ ∼ M _w^3.6, independently of the MMD. The oscillatory data were fitted with a Carreau–Yasuda equation. The resulting parameters were correlated to molecular structure. The parameter a, being a quantity for the width of the transition between the Newtonian and the non-Newtonian regime, showed a dependence on the molar mass M _w but not on M _w/M _n. The parameter λ of the Carreau-Yasuda equation was found to be the reciprocal crossover frequency for all samples with a log-Gaussian MMD. λ depends on the molar mass M _w and also on M _w/M _n.

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.     

Radial Solutions and Phase Separation in a System of Two Coupled Schrödinger Equations

We consider the nonlinear elliptic system

Oscillatory flow in microchannels

Commonly used, lumped-parameter expressions for the impedance of an incompressible viscous fluid subjected to harmonic oscillations in a channel were compared with exact expressions, based on solutions of the Navier-Stokes equations for slots and channels of circular and rectangular cross-section, and were found to differ by as much as 30% in amplitude. These differences resulted in predicted discrepancies by as much as 400% in frequency response amplitude for simple second-order systems based on size scales and frequencies encountered in microfluidic devices. These predictions were verified experimentally for rectangular microchannels and indicate that underdamped fluidic systems operating near the corner frequency of any included flow channel should be modeled with exact expressions for impedance to avoid potentially large errors in predicted behavior.List of symbols A Channel cross-sectional area (m²) - A_c Membrane area (m²) - a Rectangular duct and slot half-width or radius (m) - b Rectangular duct half-depth and slot depth (m) - C Capacitance (m³/Pa) - C - D_h Channel hydraulic diameter (m) - E Voltage (V) - f Darcy friction factor - F Force (N) - I Channel inertance (Pa s²/m³) - i - Imaginary part of a complex number - J_k Bessel function of the first kind of order k - System transfer function - K Sum of minor loss factors - k Membrane stiffness (N/m) - L Channel length (m) - n Outward unit normal vector - P Fluid pressure (Pa) - p_n - Q Volumetric flow rate (m³/s) - R Channel resistance (Pa s/m³) - Real part of a complex number - Re Reynolds number, - V Velocity (m/s) - _V Volume (m³) - w Axial component of velocity (m/s) - Harmonic amplitude of membrane centerline displacement - Fluid impedance (kg/m⁴ s) - Duct aspect ratio, b/a - ² Nondimensional frequency parameter, - Nondimensional corner frequency, - Membrane shape factor - C/C - µ Fluid dynamic viscosity (Pa s) - Fluid kinematic viscosity (m²/s) - Mass density (kg/m³) - Radian frequency - _c R_s/I_s cutoff or corner frequency - _n Undamped natural frequency - Channel shape parameter in Eqs. 29 and 30 - Damping ratio - ( )_e Exact property - ( )_s Simplified property - (^–) Spatial average - Complex quantity     

Influence of molecular parameters on the stress dependence of viscous and elastic properties of polypropylene melts in shear

The stress dependencies of the steady-state viscosity η and, particularly, that of the steady-state elastic compliance J _e of various linear isotactic polypropylenes (PP) and one long-chain branched PP are investigated using creep-recovery tests. The creep stresses applied range from 2 to 10,000 Pa. In order to discuss the stress-dependent viscosity η and elastic compliance J _e with respect to the influence of the weight average molar mass M _w and the polydispersity factor M _w/M _n the PP are characterized by SEC–MALLS. For the linear PP, linear steady-state elastic compliances J_e⁰J_{\rm e}^0 in the range of 10^− 5–10^− 3 Pa^− 1 are obtained depending on the molar mass distribution. J_e⁰J_{\rm e}^0 of the LCB-PP is distinctly higher and comes to lie at around 10^− 2 Pa^− 1. J_e⁰J_{\rm e}^0 is found to be independent of M _w but strongly dependent on polydispersity. η and J _e decrease with increasing stress. For the linear PP, J _e as a function of the stress τ is temperature independent. The higher M _w/M _n the stronger is the shear thinning of η and the more pronounced is the stress dependence of J _e. For the LCB-PP, the strongest stress dependence of η and J _e is observed. Furthermore, for all PP J _e reacts more sensitively to an increasing stress than η. A qualitative explanation for the stronger stress dependence of J _e compared to η is given by analyzing the contribution of long relaxation times to the viscosity and elasticity.     

Influence of long-chain branching on time-pressure and time-temperature shift factors for polystyrene and polyethylene

Rheological characterizations were carried out for two polystyrenes. One was a linear polymer with M _w =222,000 g/mol and M _w /M _n =2, while the other was a randomly branched polystyrene with M _w =678,000 g/mol and a broad molecular weight distribution. Experiments performed included oscillatory shear to determine the storage and loss moduli as functions of frequency and temperature, viscosity as a function of shear rate and pressure, and multi-angle light scattering to determine the radius of gyration as a function of molecular weight. The presence of branching in one sample was clearly revealed by the radius of gyration and the low-frequency portion of the complex viscosity curve. Data are also shown for three polyethylene copolymers, one (LLDPE) made using a Ziegler catalyst and two made using metallocene catalysts, one (BmPE) with and one (LmPE) without long-chain branching (LCB). While the distribution of comonomer is known to be much more uniform in LmPE than in LLDPE, the pressure shift factors were the same for these two polymers. The pressure and temperature shift factors of the two polystyrenes were identical, but, in the case of polyethylene, the presence of a small amount of LCB in the BmPE had a definite effect on the shift factors. These observations are discussed in terms of the relative roles of free volume and thermal activation in the effects of temperature and pressure.     

Asymptotic Stability of the Wave Equation on Compact Manifolds and Locally Distributed Damping: A Sharp Result

Let (M, g) be a n-dimensional ( ${n\geqq 2}Let (M, g) be a n-dimensional ( n\geqq 2{n\geqq 2}) compact Riemannian manifold with boundary where g denotes a Riemannian metric of class C ^∞. This paper is concerned with the study of the wave equation on (M, g) with locally distributed damping, described by

l utt - Dgu+ a(x) g(ut)=0,   on M×] 0,￥[ ,u=0 on ?M ×] 0,￥[, \left. \begin{array}{l} u_{tt} - \Delta_{{\bf g}}u+ a(x)\,g(u_{t})=0,\quad\hbox{on\ \thinspace}{M}\times \left] 0,\infty\right[ ,u=0\,\hbox{on}\,\partial M \times \left] 0,\infty \right[, \end{array} \right.      8. Forced convection laminar film condensation on an inclined elliptical tube in the absence of gravity      M. Mosaad 《Heat and Mass Transfer》2005,41(11):953-960 In this paper, the problem of forced convection dominated laminar film condensation, in the absence of gravity, on an inclined elliptical tube is investigated theoretically. In this analysis, the interfacial vapour shear stress is modelled following Shekriladze approach. By employing the method of characteristics, expressions are analytically derived for calculating the local film thickness as well as the local and mean Nusselt numbers. The results show that the mean Nusselt number, enhances with the increase in the tube ellipticity. For the practical ellipticity range: 0.8 e 0.92, this enhancement is found from 7 to 14% compared to a circular tube with the same length and equivalent condensation surface area. M. MosaadEmail:       9. Linear viscoelasticity,simple and planar melt extension of linear polybutadienes with bimodal molar mass distributions      Dr. L. Berger  Prof. Dr. J. Meissner 《Rheologica Acta》1992,31(1):63-74 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.      10. Re-examination of terminal relaxation behavior of high-molecular-weight ring polystyrene melts      Yuya?Doi  Atsushi?Matsumoto  Tadashi?Inoue  Takuro?Iwamoto  Atsushi?TakanoEmail author  Yushu?Matsushita  Yoshiaki?Takahashi  Hiroshi?WatanabeEmail author 《Rheologica Acta》2017,56(6):567-581 For high-molecular-weight (M) ring polymers with low contamination of linear chains, recent viscoelastic tests revealed broad terminal relaxation associated with no clear entanglement plateau. This relaxation behavior is qualitatively similar to that deduced from molecular models (double-folded lattice-animal model and the fractal loopy globule model) for entangled ring polymers, but quantitatively important differences are also noted: For example, the full terminal relaxation of those polymers is slower than the model prediction. This study re-examined the viscoelastic data of entangled high-M ring polystyrene (PS) samples (coded as R-240; M = 244×103) specifically for two points: the purity of the ring samples after the viscoelastic tests and the molecular origin of the stress. For the first point, the R-240 samples contaminated with linear chains at low but different levels were prepared by tuning either the purification efficiency or the retention time of the sample at high temperature (T) before/during the viscoelastic test. The fraction w L of the linear contaminant, determined after the viscoelastic measurement, was ranging from 0.7 to 4.9%, and the extrapolation of the modulus data to w L = 0 gave the data for the ideally pure ring melt. This pure ring melt exhibited broad terminal relaxation that started faster but completed slower compared to the model prediction, indicating that the ring relaxation is not well described by the current model(s) even in the absence of linear contaminant. For the second point, dynamic birefringence measurements were conducted for the R-240 samples with w L = 4.6 and 1.0%. These samples obeyed the stress-optical rule, and their stress-optical coefficient was indistinguishable from that for linear PS samples, revealing that the stress of the ring PS chains reflects the orientational anisotropy of the chains (as is the case also for linear chains). The relaxation behavior of pure ring PS melt is discussed on the basis of these findings, with the focus being placed on the ring-ring threading not considered in the models.      11. A numerical study of various rheological polydispersity measures   总被引：1，自引：0，他引：1   P. A. M. Steeman 《Rheologica Acta》1998,37(6):583-592 Model calculations were performed in order to investigate the sensitivity of various rheological polydispersity parameters for variations in the moments of the molar mass distribution (MMD) of linear polymers. Molar mass distributions were generated with the Gaussian and the Generalised exponential distribution functions, using a fixed weight average molar mass M w and variable M w /M n and M z /M n . Assuming linear entangled polymeric chains, the linear viscoelastic properties were predicted by calculating the stress relaxation modulus of the consecutive monodisperse fractions with the BSW relaxation time spectrum and blending these curves with the double reptation blending rule. BSW relaxation parameters appropriate for polypropylene were used.  It was found that both the zero-shear viscosity and the so-called cross-over frequency, at which and are equal, depend mostly on M w but also significantly on both M w /M n and M z /M w . By contrast, the steady-state compliance depends mainly on M z /M w , its functional dependence on moments of the MMD being best described by the Ferry equation.  None of the polydispersity parameters PI (from the modulus cross-over), MODSEP (the modulus separation) or PDR (from the shape of the flow curve), as introduced in literature depends solely on the polydispersity M w /M n . PI is the most sensitive indicator for this purpose. Finally, the parameters ER ( at a fixed low value of , MODSEP en DRI (from the shape of the flow curve) are shown to be good indicators for the weight (M z /M w ) of the high molar mass tail of the molar mass distribution. Received: 5 May 1998 Accepted: 30 July 1998      12. On the existence of the derivative of the volume average      Jiří Mls 《Transport in Porous Media》1987,2(6):615-621 A general theorem on the derivative of the volume average is formulated and proved. Conditions for the existence of the derivative are presented and discussed. This is done in order to give a better base to the theory of spatial averaging.Latin Letters E 3 three-dimensional vector space over the field of real numbers - K, K(x) averaging domain - G, G w, Gs open sets in E 3; components of the two-phase system - C 1(G) the set of functions 1-times continuously differentiable in G - W1/2(G) Sobolev space - V volume of the domain K - f function defined in G, G w - K infi sup* (x), K infi sup– (x) special parts of K(x) Greek Letters boundary of G, G w, Gs; w-s interface - ij Kronecker delta - v unit outward normal of G, G w - j j-dimensional Lebesgue measure Other M closure of a set M in the metric space E 3 - f phase average of f for the w-phase - (u, v) scalar product of u, v in E 3 - one-sided derivatives      13. Transient shear and elongational behavior of blends of PET with a LCP      Lucineide Balbino da Silva  Marcelo Massayoshi Ueki  Marcelo Farah  Vitor Manuel Coelho Barroso  João Manuel Luis Lopes Maia  Rosario Elida Suman Bretas 《Rheologica Acta》2006,45(3):268-280 Blends of polyethylene terephthalate (PET) with a liquid crystalline polymer (LCP) and a compatibilizer were produced by twin screw extrusion and injection molding. Transesterification and compatibilization studies were made in a torque rheometer. The morphology of the injection-molded plaques was studied by scanning electron microscopy. The blends shear growth function was measured in a cone and plate rheometer. The elongational growth function was measured in a modified rotational rheometer. Transesterification was observed in the PET/LCP/compatibilizer 95/5/0 blend. The injection-molded plaques displayed the usual “skin-core” morphology. All the blends were highly shear-thinning, even at low shear rates; thus, a zero-shear viscosity could not be calculated. The compatibilized blend had the highest shear viscosity of all the blends, confirming the strong PET/LCP interphase and the effectiveness of the compatibilizing agent. On the other hand, the 90/10/0 blend had the lowest shear viscosity. All the blends showed strain softening behavior, similar to the PET. The 90/10/0 blend had the highest elongational growth function, while the 95/5/0 had the lowest. The compatibilized blend had an intermediate behavior between both blends.      14. Rheologische Messungen an Polypropylenschmelzen      H. P. Frank 《Rheologica Acta》1968,7(4):344-348 Zusammenfassung Es wurden Fließkurven von Polypropylenschmelzen bei 190 °C in einem Schergefällebereich von etwa 10–1 bis 102 sec–1 aufgenommen, wobei drei Serien von Polymerproben herangezogen wurden: eine Serie PP 1–7 von isotaktischen Propylenpolymerisaten mit einer Molekulargewichtsverteilung, die durch¯M w /¯M n 4–6 gekennzeichnet war; eine zweite Serie PP 1d–5d von abgebauten Polymerproben mit¯M w /¯M n 2–3; eine dritte Serie PP 1m–3m, die aus Mischungen von Polypropylenen bestand und die durch¯M w /¯M n >10 gekennzeichnet war. Es wurde der Einfluß des Molekulargewichts und der Molekulargewichtsverteilung auf die Fließkurven untersucht, insbesondere der Einfluß auf den Exponentenn in derOstwald-de Waele-Gleichung=KD n . Es wurde versucht, die Ergebnisse durch die universaleVinogradov-Viskositätsfunktion darzustellen, wobei sich aber sehr wesentliche Abweichungen infolge Variation des Molekulargewichts und der Molekulargewichtsverteilung ergaben. Summary Flow curves of molten polypropylene were determined at 190 °C over a shear gradient range of about l0–1 to 102 sec–1. Three series of polymer samples have been used: one series PP 1–7 of isotactic propylene polymerization products with a molecular weight distribution characterized by¯M w /¯M n 4–6, another series of degraded polymer samples PP 1d–5d with¯M w /¯M n 2–3, and still another series of polymer mixtures PP 1m–3m with¯M w /¯M n >10. The influence of molecular weight and molecular weight distribution on the flow curves was investigated, particularly on the exponentn in theOstwald-de Waele-power law=KD n . An attempt to plot the results by the universalVinogradov-viscosity function produced marked deviations due to variation of molecular weight and molecular weight distribution. Vorgetragen auf der Arbeitstagung der Sektion Rheologie des Vereins Österreichischer Chemiker am 17. November 1967 in Graz. Mitteilung I, II: Rheol. Acta5, 89 (1966); ibid. im Druck [Messung und Interpretation von Filmkurven].      15. Uniaxial elongational behavior of poly(vinyl chloride) physical gel      Yuji Aoki  Kentaro Hirayama  Koji Kikuchi  Masataka Sugimoto  Kiyohito Koyama 《Rheologica Acta》2010,49(10):1071-1076 A poly(vinyl chloride) (PVC,  Mw = 102×103)(\mbox{PVC,}\;{\rm M}_{\rm w} =102\times 10^3) di-octyl phthalate (DOP) gel with PVC content of 20 wt.% was prepared by a solvent evaporation method. The dynamic viscoelsticity and elongational viscosity of the PVC/DOP gel were measured at various temperatures. The gel exhibited a typical sol–gel transition behavior with elevating temperature. The critical gel temperature (Tgel) characterized with a power–law relationship between the storage and loss moduli, G′ and G″, and frequency ω, G￠=G￠￠/tan  ( np/2 ) μ wn{G}^\prime={G}^{\prime\prime}{\rm /tan}\;\left( {{n}\pi {\rm /2}} \right)\propto \omega ^{n}, was observed to be 152°C. The elongational viscosity of the gel was measured below the Tgel. The gel exhibited strong strain hardening. Elongational viscosity against strain plot was independent of strain rate. This finding is different from the elongational viscosity behavior of linear polymer solutions and melts. The stress–strain relations were expressed by the neo-Hookean model at high temperature (135°C) near the Tgel. However, the stress–strain curves were deviated from the neo-Hookean model at smaller strain with decreasing temperature. These results indicated that this physical gel behaves as the neo-Hookean model at low cross-linking point, and is deviated from the neo-Hookean model with increasing of the PVC crystallites worked as the cross-linking junctions.      16. Effect of temperature and molecular weight on the interfacial tension of PS/PDMS blends      Girma Biresaw  Craig J. Carriere  Robert L. Sammler 《Rheologica Acta》2003,42(1-2):142-147 The imbedded-fiber retraction (IFR) method was used to study the effect of temperature and PDMS molecular weight on the interfacial tension of PS/PDMS blends. The interfacial tension decreased with increasing temperature and analysis of the temperature dependence using a simple linear fit gave –dγ/dT value of 0.058±0.010 dyn/cm-deg. Reported –dγ/dT values of PS/PDMS blends are highly dependent on the molecular weights of the polymers and can have values that are <0, 0, or >0. Our interfacial tension values were independent of the molecular weight of PDMS and this was attributed to the molecular weights studied here being well above the entanglement values of both polymers. However, analysis of interfacial tension data from this work and the literature showed the following empirical relationship between apparent blend molecular weight, Mb, and interfacial tension of PS/PDMS blends with a correlation of 0.94: γ12=γ0+k2Mb (–2/3), where γ0=7.3±0.3 dyn/cm; k2=–517±41 (dyn/cm)(g/mol)2/3.      17. Measurement of temperature field of a Rayleigh-Bénard convection using two-color laser-induced fluorescence      J. Sakakibara  R. J. Adrian 《Experiments in fluids》2004,37(3):331-340       18. Viscoelastic properties of ultra-high viscosity alginates      Henning Storz  Ulrich Zimmermann  Heiko Zimmermann  Werner-Michael Kulicke 《Rheologica Acta》2010,49(2):155-167 Ultra-high viscosity alginates were extracted from the brown seaweeds Lessonia nigrescens (UHVN, containing 61% mannuronate (M) and 2% guluronate (G)) and Lessonia trabeculata (UHVT, containing 22% M and 78% G). The viscoelastic behavior of the aqueous solutions of these alginates was determined in shear flow in terms of the shear stress σ 21, the first normal stress difference N 1, and the shear viscosity η in isotonic NaCl solutions (0.154 mol/L) at T = 298 K in dependence of the shear rate [(g)\dot]\dot{\gamma} for solutions of varying concentrations and molar masses (3–10 × 105 g/mol, homologous series was prepared by ultrasonic degradation). Data obtained in small-amplitude oscillatory shear (SAOS) experiments obey the Cox–Merz rule. For comparison, a commercial alginate with intermediate chemical composition was additionally characterized. Particulate substances which are omnipresent in most alginates influenced the determination of the material functions at low shear rates. We have calculated structure–property relationships for the prediction of the viscosity yield, e.g., η–M w–c–[(g)\dot]\dot{\gamma} for the Newtonian and non-Newtonian region. For the highest molar masses and concentrations, the elasticity yield in terms of N 1 could be determined. In addition, the extensional flow behavior of the alginates was measured using capillary breakup extensional rheometry. The results demonstrate that even samples with the same average molar mass but different molar mass distributions can be differentiated in contrast to shear flow or SAOS experiments.      19. Existence of Solutions with the Prescribed Flux of the Navier–Stokes System in an Infinite Cylinder      K. Pileckas 《Journal of Mathematical Fluid Mechanics》2006,8(4):542-563 The existence and uniqueness of a solution to the nonstationary Navier–Stokes system having a prescribed flux in an infinite cylinder is proved. We assume that the initial data and the external forces do not depend on x3 and find the solution (u, p) having the following form where x′  =  (x1, x2). Such solution generalize the nonstationary Poiseuille solutions.      20. Linear and branched poly(butyleneterephthalate): Activation energy for melt flow      A. Munari  F. Pilati  G. Pezzin 《Rheologica Acta》1985,24(5):534-536 Poly(butyleneterephthalate) (PBTP) samples of different molecular weights, both linear and branched, were synthetized by mass polymerization and studied in the molten state with a melt-flow-index apparatus at different temperatures in the range 245–270 °C. In our experimental conditions ( 20 s–1) the behaviour of PBTP samples was Newtonian, as reported previously. The flow activation energyE a,0 was found to increase with degree of branching: typicallyE a,0 was about 47 and 63–79 kJ/mol for linear and branched polymers respectively.Presented in part at the IX. International Congress on Rheology, Acapulco, Mexico, October 8–13, 1984

Forced convection laminar film condensation on an inclined elliptical tube in the absence of gravity

In this paper, the problem of forced convection dominated laminar film condensation, in the absence of gravity, on an inclined elliptical tube is investigated theoretically. In this analysis, the interfacial vapour shear stress is modelled following Shekriladze approach. By employing the method of characteristics, expressions are analytically derived for calculating the local film thickness as well as the local and mean Nusselt numbers. The results show that the mean Nusselt number, enhances with the increase in the tube ellipticity. For the practical ellipticity range: 0.8 e 0.92, this enhancement is found from 7 to 14% compared to a circular tube with the same length and equivalent condensation surface area.

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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.     

Re-examination of terminal relaxation behavior of high-molecular-weight ring polystyrene melts

For high-molecular-weight (M) ring polymers with low contamination of linear chains, recent viscoelastic tests revealed broad terminal relaxation associated with no clear entanglement plateau. This relaxation behavior is qualitatively similar to that deduced from molecular models (double-folded lattice-animal model and the fractal loopy globule model) for entangled ring polymers, but quantitatively important differences are also noted: For example, the full terminal relaxation of those polymers is slower than the model prediction. This study re-examined the viscoelastic data of entangled high-M ring polystyrene (PS) samples (coded as R-240; M = 244×10³) specifically for two points: the purity of the ring samples after the viscoelastic tests and the molecular origin of the stress. For the first point, the R-240 samples contaminated with linear chains at low but different levels were prepared by tuning either the purification efficiency or the retention time of the sample at high temperature (T) before/during the viscoelastic test. The fraction w _L of the linear contaminant, determined after the viscoelastic measurement, was ranging from 0.7 to 4.9%, and the extrapolation of the modulus data to w _L = 0 gave the data for the ideally pure ring melt. This pure ring melt exhibited broad terminal relaxation that started faster but completed slower compared to the model prediction, indicating that the ring relaxation is not well described by the current model(s) even in the absence of linear contaminant. For the second point, dynamic birefringence measurements were conducted for the R-240 samples with w _L = 4.6 and 1.0%. These samples obeyed the stress-optical rule, and their stress-optical coefficient was indistinguishable from that for linear PS samples, revealing that the stress of the ring PS chains reflects the orientational anisotropy of the chains (as is the case also for linear chains). The relaxation behavior of pure ring PS melt is discussed on the basis of these findings, with the focus being placed on the ring-ring threading not considered in the models.     

A numerical study of various rheological polydispersity measures

Model calculations were performed in order to investigate the sensitivity of various rheological polydispersity parameters for variations in the moments of the molar mass distribution (MMD) of linear polymers. Molar mass distributions were generated with the Gaussian and the Generalised exponential distribution functions, using a fixed weight average molar mass M _w and variable M _w /M _n and M _z /M _n . Assuming linear entangled polymeric chains, the linear viscoelastic properties were predicted by calculating the stress relaxation modulus of the consecutive monodisperse fractions with the BSW relaxation time spectrum and blending these curves with the double reptation blending rule. BSW relaxation parameters appropriate for polypropylene were used.  It was found that both the zero-shear viscosity and the so-called cross-over frequency, at which and are equal, depend mostly on M _w but also significantly on both M _w /M _n and M _z /M _w . By contrast, the steady-state compliance depends mainly on M _z /M _w , its functional dependence on moments of the MMD being best described by the Ferry equation.  None of the polydispersity parameters PI (from the modulus cross-over), MODSEP (the modulus separation) or PDR (from the shape of the flow curve), as introduced in literature depends solely on the polydispersity M _w /M _n . PI is the most sensitive indicator for this purpose. Finally, the parameters ER ( at a fixed low value of , MODSEP en DRI (from the shape of the flow curve) are shown to be good indicators for the weight (M _z /M _w ) of the high molar mass tail of the molar mass distribution. Received: 5 May 1998 Accepted: 30 July 1998     

On the existence of the derivative of the volume average

A general theorem on the derivative of the volume average is formulated and proved. Conditions for the existence of the derivative are presented and discussed. This is done in order to give a better base to the theory of spatial averaging.Latin Letters E ₃ three-dimensional vector space over the field of real numbers - K, K(x) averaging domain - G, G _w, G_s open sets in E ₃; components of the two-phase system - C ¹(G) the set of functions 1-times continuously differentiable in G - W1/2(G) Sobolev space - V volume of the domain K - f function defined in G, G _w - K _infi ^sup* (x), K _infi ^sup– (x) special parts of K(x) Greek Letters boundary of G, G _w, G_s; w-s interface - _ij Kronecker delta - v unit outward normal of G, G _w - _j j-dimensional Lebesgue measure Other M closure of a set M in the metric space E ₃ - f phase average of f for the w-phase - (u, v) scalar product of u, v in E ₃ - one-sided derivatives     

Transient shear and elongational behavior of blends of PET with a LCP

Blends of polyethylene terephthalate (PET) with a liquid crystalline polymer (LCP) and a compatibilizer were produced by twin screw extrusion and injection molding. Transesterification and compatibilization studies were made in a torque rheometer. The morphology of the injection-molded plaques was studied by scanning electron microscopy. The blends shear growth function was measured in a cone and plate rheometer. The elongational growth function was measured in a modified rotational rheometer. Transesterification was observed in the PET/LCP/compatibilizer 95/5/0 blend. The injection-molded plaques displayed the usual “skin-core” morphology. All the blends were highly shear-thinning, even at low shear rates; thus, a zero-shear viscosity could not be calculated. The compatibilized blend had the highest shear viscosity of all the blends, confirming the strong PET/LCP interphase and the effectiveness of the compatibilizing agent. On the other hand, the 90/10/0 blend had the lowest shear viscosity. All the blends showed strain softening behavior, similar to the PET. The 90/10/0 blend had the highest elongational growth function, while the 95/5/0 had the lowest. The compatibilized blend had an intermediate behavior between both blends.     

Rheologische Messungen an Polypropylenschmelzen

Zusammenfassung Es wurden Fließkurven von Polypropylenschmelzen bei 190 °C in einem Schergefällebereich von etwa 10^–1 bis 10² sec^–1 aufgenommen, wobei drei Serien von Polymerproben herangezogen wurden: eine Serie PP 1–7 von isotaktischen Propylenpolymerisaten mit einer Molekulargewichtsverteilung, die durch¯M _w /¯M _n 4–6 gekennzeichnet war; eine zweite Serie PP 1d–5d von abgebauten Polymerproben mit¯M _w /¯M _n 2–3; eine dritte Serie PP 1m–3m, die aus Mischungen von Polypropylenen bestand und die durch¯M _w /¯M _n >10 gekennzeichnet war. Es wurde der Einfluß des Molekulargewichts und der Molekulargewichtsverteilung auf die Fließkurven untersucht, insbesondere der Einfluß auf den Exponentenn in derOstwald-de Waele-Gleichung=KD ⁿ . Es wurde versucht, die Ergebnisse durch die universaleVinogradov-Viskositätsfunktion darzustellen, wobei sich aber sehr wesentliche Abweichungen infolge Variation des Molekulargewichts und der Molekulargewichtsverteilung ergaben.

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Summary Flow curves of molten polypropylene were determined at 190 °C over a shear gradient range of about l0^–1 to 10² sec^–1. Three series of polymer samples have been used: one series PP 1–7 of isotactic propylene polymerization products with a molecular weight distribution characterized by¯M _w /¯M _n 4–6, another series of degraded polymer samples PP 1d–5d with¯M _w /¯M _n 2–3, and still another series of polymer mixtures PP 1m–3m with¯M _w /¯M _n >10. The influence of molecular weight and molecular weight distribution on the flow curves was investigated, particularly on the exponentn in theOstwald-de Waele-power law=KD ⁿ . An attempt to plot the results by the universalVinogradov-viscosity function produced marked deviations due to variation of molecular weight and molecular weight distribution.

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Vorgetragen auf der Arbeitstagung der Sektion Rheologie des Vereins Österreichischer Chemiker am 17. November 1967 in Graz. Mitteilung I, II: Rheol. Acta5, 89 (1966); ibid. im Druck [Messung und Interpretation von Filmkurven].     

Uniaxial elongational behavior of poly(vinyl chloride) physical gel

A poly(vinyl chloride) (PVC,  M_w = 102×10³)(\mbox{PVC,}\;{\rm M}_{\rm w} =102\times 10^3) di-octyl phthalate (DOP) gel with PVC content of 20 wt.% was prepared by a solvent evaporation method. The dynamic viscoelsticity and elongational viscosity of the PVC/DOP gel were measured at various temperatures. The gel exhibited a typical sol–gel transition behavior with elevating temperature. The critical gel temperature (T_gel) characterized with a power–law relationship between the storage and loss moduli, G^′ and G^″, and frequency ω, G￠=G^￠￠/tan  ( np/2 ) μ wⁿ{G}^\prime={G}^{\prime\prime}{\rm /tan}\;\left( {{n}\pi {\rm /2}} \right)\propto \omega ^{n}, was observed to be 152°C. The elongational viscosity of the gel was measured below the T_gel. The gel exhibited strong strain hardening. Elongational viscosity against strain plot was independent of strain rate. This finding is different from the elongational viscosity behavior of linear polymer solutions and melts. The stress–strain relations were expressed by the neo-Hookean model at high temperature (135°C) near the T_gel. However, the stress–strain curves were deviated from the neo-Hookean model at smaller strain with decreasing temperature. These results indicated that this physical gel behaves as the neo-Hookean model at low cross-linking point, and is deviated from the neo-Hookean model with increasing of the PVC crystallites worked as the cross-linking junctions.     

Effect of temperature and molecular weight on the interfacial tension of PS/PDMS blends

The imbedded-fiber retraction (IFR) method was used to study the effect of temperature and PDMS molecular weight on the interfacial tension of PS/PDMS blends. The interfacial tension decreased with increasing temperature and analysis of the temperature dependence using a simple linear fit gave –dγ/dT value of 0.058±0.010 dyn/cm-deg. Reported –dγ/dT values of PS/PDMS blends are highly dependent on the molecular weights of the polymers and can have values that are <0, 0, or >0. Our interfacial tension values were independent of the molecular weight of PDMS and this was attributed to the molecular weights studied here being well above the entanglement values of both polymers. However, analysis of interfacial tension data from this work and the literature showed the following empirical relationship between apparent blend molecular weight, M_b, and interfacial tension of PS/PDMS blends with a correlation of 0.94: γ₁₂=γ₀+k₂M_b ^(–2/3), where γ₀=7.3±0.3 dyn/cm; k₂=–517±41 (dyn/cm)(g/mol)^2/3.     

Viscoelastic properties of ultra-high viscosity alginates

Ultra-high viscosity alginates were extracted from the brown seaweeds Lessonia nigrescens (UHV_N, containing 61% mannuronate (M) and 2% guluronate (G)) and Lessonia trabeculata (UHV_T, containing 22% M and 78% G). The viscoelastic behavior of the aqueous solutions of these alginates was determined in shear flow in terms of the shear stress σ ₂₁, the first normal stress difference N ₁, and the shear viscosity η in isotonic NaCl solutions (0.154 mol/L) at T = 298 K in dependence of the shear rate [(g)\dot]\dot{\gamma} for solutions of varying concentrations and molar masses (3–10 × 10⁵ g/mol, homologous series was prepared by ultrasonic degradation). Data obtained in small-amplitude oscillatory shear (SAOS) experiments obey the Cox–Merz rule. For comparison, a commercial alginate with intermediate chemical composition was additionally characterized. Particulate substances which are omnipresent in most alginates influenced the determination of the material functions at low shear rates. We have calculated structure–property relationships for the prediction of the viscosity yield, e.g., η–M _w–c–[(g)\dot]\dot{\gamma} for the Newtonian and non-Newtonian region. For the highest molar masses and concentrations, the elasticity yield in terms of N ₁ could be determined. In addition, the extensional flow behavior of the alginates was measured using capillary breakup extensional rheometry. The results demonstrate that even samples with the same average molar mass but different molar mass distributions can be differentiated in contrast to shear flow or SAOS experiments.     

Existence of Solutions with the Prescribed Flux of the Navier–Stokes System in an Infinite Cylinder

The existence and uniqueness of a solution to the nonstationary Navier–Stokes system having a prescribed flux in an infinite cylinder is proved. We assume that the initial data and the external forces do not depend on x₃ and find the solution (u, p) having the following form

Linear and branched poly(butyleneterephthalate): Activation energy for melt flow

Poly(butyleneterephthalate) (PBTP) samples of different molecular weights, both linear and branched, were synthetized by mass polymerization and studied in the molten state with a melt-flow-index apparatus at different temperatures in the range 245–270 °C. In our experimental conditions ( 20 s^–1) the behaviour of PBTP samples was Newtonian, as reported previously. The flow activation energyE _a,0 was found to increase with degree of branching: typicallyE _a,0 was about 47 and 63–79 kJ/mol for linear and branched polymers respectively.Presented in part at the IX. International Congress on Rheology, Acapulco, Mexico, October 8–13, 1984