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. 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

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.     

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